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Find x****.
SOLUTION: In a 45°-45°-90° triangle, the legs are congruent (
= ) and the length of the hypotenuse h is times
the length of a leg. Therefore, since the side length ( ) is 5, then ANSWER:
SOLUTION: In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times the length of a leg. Therefore, since the hypotenuse (h) is 14, then Solve for x. ANSWER:
SOLUTION:
In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times the length of a leg. Therefore, since , then. Simplify: ANSWER: 22
Find x and y****.
SOLUTION: In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( 2s ) and the length of the longer leg is times the length of the shorter leg ( ). The length of the hypotenuse is the shorter leg is y , and the longer leg is x. Therefore, Solve for y : Substitute and solve for x: ANSWER: ;
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s (2 s ) and the length of the longer leg is times the length of the shorter leg ( ). The length of the hypotenuse is x , the shorter leg is 7, and the longer leg is y. Therefore, ANSWER: ;
SENSE-MAKING Find x****.
SOLUTION: In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times the length of a leg. Therefore, since the hypotenuse is 16 and the legs are x , then Solve for x. ANSWER:
SOLUTION:
In a 45°-45°-90° triangle, the legs are congruent ( = ) and the length of the hypotenuse h is times the length of a leg. Since the hypotenuse is 15 and the legs are x , then Solve for x. ANSWER: or
SOLUTION:
In a 45°-45°-90° triangle, the legs l are congruent and the length of the hypotenuse h is times the length of a leg. Since the the legs are , then the hypotenuse is . Simplify: ANSWER: 34
SOLUTION:
In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Therefore, since the legs are , the hypotenuse is Simplify: ANSWER:
SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Therefore, since the legs are 19.5, then the hypotenuse would be ANSWER:
- Find the length of the hypotenuse of a - - triangle with a leg length of 8 centimeters. SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Therefore, since the legs are 8, the hypotenuse would be ANSWER: or 11.3 cm Find x and y****.
- SOLUTION: In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s (h=2s) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is y , the shorter leg is x , and the longer leg is.
Therefore,.
Solve for x :
Then , to find the hypotenuse, ANSWER: x = 8; y = 16
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is 7 , the shorter leg is x , and the longer leg is.
Therefore,.
Solve for x :
Then , to find the hypotenuse, ANSWER: x = 10; y = 20
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is 15, the shorter leg is y , and the longer leg is x.
Therefore,.
Solve for y :
Substitute and solve for x : ANSWER: ;
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is 17, the shorter leg is y , and the longer leg is x.
Therefore,.
Solve for y :
Substitute and solve for x : ANSWER: ;
- An equilateral triangle has an altitude length of 18 feet. Determine the length of a side of the triangle. SOLUTION: Let x be the length of each side of the equilateral triangle. The altitude from one vertex to the opposite side divides the equilateral triangle into two 30°-60°-90° triangles. In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is x , the shorter leg is , and the longer leg is 18.
Therefore,.
Solve for x :
ANSWER:
or 20.8 ft
- Find the length of the side of an equilateral triangle that has an altitude length of 24 feet. SOLUTION: Let x be the length of each side of the equilateral triangle. The altitude from one vertex to the opposite side divides the equilateral triangle into two 30°-60°-90° triangles. In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The length of the hypotenuse is x , the shorter leg is , and the longer leg is 24.
Therefore,.
Solve for x :
ANSWER:
or 27.7 ft
- MODELING Refer to the beginning of the lesson. Each highlighter is an equilateral triangle with 9 cm sides. Will the highlighter fit in a 10 cm by 7 cm rectangular box? Explain. SOLUTION: Find the height of the highlighter. The altitude from one vertex to the opposite side divides the equilateral triangle into two 30°-60°-90° triangles. Let x be the height of the triangle. Use special right triangles to find the height, which is the longer side of a 30°-60°-90° triangle.
The hypotenuse of this 30°-60°-90° triangle is 9, the
shorter leg is , which makes the height , which is approximately 7.8 cm. The height of the box is only 7 cm. and the height of the highlighter is about 7.8 cm., so it will not fit. ANSWER: No; sample answer: The height of the box is only 7 cm. and the height of the highlighter is about 7.8 cm., so it will not fit.
- EVENT PLANNING Grace is having a party, and she wants to decorate the gable of the house as shown. The gable is an isosceles right triangle and she knows that the height of the gable is 8 feet. What length of lights will she need to cover the gable below the roof line? SOLUTION: The gable is a 45°-45°-90° triangle. The altitude again
divides it into two 45°-45°-90° triangles. The length of
the leg of each triangle is 8 feet. In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. If , then. Since there are two hypotenuses that have to be decorated, the total length is ANSWER: 22.6 ft
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. In one of the 30-60-90 triangles in this figure, the length of the hypotenuse is , the shorter leg is s , and the longer leg is x.
Therefore,.
Solve for s :
Substitute and solve for x :
In a different 30-60-90 triangles in this figure, the length of the shorter leg is y and the longer leg is.
Therefore,.
Solve for y : ANSWER: x = 3; y = 1
SOLUTION:
In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Since y is the hypotenuse of a 45°-45°-90° triangle whose each leg measures , then Since x is a leg of a 45°-45°-90° triangle whose hypotenuse measures , then Solve for x : ANSWER: x = 5; y = 10
SOLUTION:
In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. In one of the 30-60-90 triangles in this figure, the length of the hypotenuse is x , the shorter leg is , and the longer leg is 9.
Therefore,.
In a different 30-60-90 triangles in this figure, the length of the shorter leg is y , and the longer leg is .
Therefore,.
ANSWER:
; y = 3
SOLUTION: The diagonal of a square divides it into two 45°-45°-90°. Therefore, x = 45°. In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Therefore, since the legs are 12, then the hypotenuse would be ANSWER: x = 45 ;
- QUILTS The quilt block shown is made up of a square and four isosceles right triangles. What is the value of x? What is the side length of the entire quilt block? SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. Since the hypotenuse of the triangles formed by the diagonals of this square are each , then
Solve for x :
Therefore, x = 6 inches. Here, x is half the length of each side of the entire quilt block. Therefore, the length of each side of the entire quilt block is 12 inches. ANSWER: 6 in.; 12 in.
SOLUTION:
In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg or. x is the length of each leg of a 45°-45°-90° triangle whose hypotenuse measures 18 units, therefore the hypotenuse would be Solve for x : In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s ( h = 2 s ) and the length of the longer leg is times the length of the shorter leg. The hypotenuse is z , the longer leg is 18, and the shorter leg is y. Therefore,. Solve for y : Substitute and solve for z : ANSWER:
- Each triangle in the figure is a - - triangle. Find x. SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. is the hypotenuse of a 45°-45°-90° triangle whose each leg measures x. Therefore, . is the hypotenuse of a 45°-45°-90° triangle whose each leg measures Therefore, is the hypotenuse of a 45°-45°-90° triangle whose each leg measures 2 x. Therefore, is the hypotenuse of a 45°-45°-90° triangle whose each leg measures Therefore,
Since FA = 6 units, then 4 x = 6 and x =. ANSWER:
- MODELING The dump truck shown has a 15-foot bed length. What is the height of the bed h when angle x is 30°? 45°? 60°? SOLUTION: In a 45°-45°-90° triangle, the legs are congruent and the length of the hypotenuse h is times the length of a leg. In a 30°-60°-90° triangle, the length of the hypotenuse h is 2 times the length of the shorter leg s , and the length of the longer leg is times the length of the shorter leg. When x is 30°, h is the shorter leg of a 30°-60°-90° triangle whose hypotenuse is 15 ft. Therefore, When x is 45°, h is the length of each leg of a 45°-45°-90° triangle whose hypotenuse is 15 ft. Therefore, Solve for h : When x is 60°, h is the longer leg of a 30°-60°-90° triangle whose hypotenuse is 15 ft. The shorter leg is half the hypotenuse. Therefore, ANSWER: 7.5 ft; 10.6 ft; 13.0 ft
- COORDINATE GEOMETRY is a -
- triangle with. Find the coordinates of E in Quadrant III for F (–3 , –4) and G (–3 , 2). is the longer leg. SOLUTION: The side is the longer leg of the 30°-60°-90° triangle and it is 6 units long and Solve for EF : Find the point E in the third quadrant, units away (approximately 3.5 units) from F such that Therefore, the coordinates of E is ( , –4). ANSWER: ( , –4)
- COORDINATE GEOMETRY is a -
- triangle with right angle K. Find the coordinates of L in Quadrant IV for J (–3 , 5) and K (– 3 , –2). SOLUTION: The side is one of the congruent legs of the right triangle and it is 7 units. Therefore, the point L is also 7 units away from K. Find the point L in the Quadrant IV, 7 units away from K, such that Therefore, the coordinates of L is (4, –2). ANSWER: (4, –2)
- EVENT PLANNING Eva has reserved a gazebo at a local park for a party. She wants to be sure that there will be enough space for her 12 guests to be in the gazebo at the same time. She wants to allow 8 square feet of area for each guest. If the floor of the gazebo is a regular hexagon and each side is 7 feet , will there be enough room for Eva and her friends? Explain. (Hint: Use the Polygon Interior Angle Sum Theorem and the properties of special right triangles.)
SOLUTION:
A regular hexagon can be divided into a rectangle and four congruent right angles as shown. The length of the hypotenuse of each triangle is 7 ft. By the Polygon Interior Angle Theorem, the sum of the interior angles of a hexagon is (6 – 2)180 = 720. Since the hexagon is a regular hexagon, each angle is equal to Therefore, each triangle in the diagram is a 30°-60°-90° triangle, and the lengths of the shorter and longer legs of the triangle are The total area is the sum of the four congruent triangles and the rectangle of sides Therefore, total area is When planning a party with a stand-up buffet, a host should allow 8 square feet of area for each guest. Divide the area by 8 to find the number of guests that can be accommodated in the gazebo. So, the gazebo can accommodate about 16 guests. With Eva and her friends, there are a total of 13 at the party, so they will all fit. ANSWER: Yes; sample answer: The gazebo is about 127 ft², which will accommodate 16 people. With Eva and her friends, there are a total of 13 at the party, so they will all fit.
- MULTIPLE REPRESENTATIONS In this problem , you will investigate ratios in right triangles. a. Geometric Draw three similar right triangles with a angle. Label one triangle ABC where angle A is the right angle and B is the angle. Label a second triangle MNP where M is the right angle and N is the angle. Label the third triangle XYZ where X is the right angle and Y is the angle. b. Tabular Copy and complete the table below. c. Verbal Make a conjecture about the ratio of the
