Transcript SPECIAL RIGHT TRIANGLES
A B
RIGHT TRIANGLES
C
A² + B² = C² “SIDE A AND SIDE B ALWAYS MEET AT THE RIGHT ANGLE”
SPECIAL RIGHT TRIANGLES
45 ° -45 ° -90 °
30 ° -60 ° -90 °
30 °- 60 °- 90° TRIANGLES
• • • • THESE SPECIAL RIGHT TRIANGLES HAVE A 90 ° ANGLE, AND MUST HAVE A 30 ° ANGLE AND A 60 ° ANGLE.
THE SIDE DIRECTLY ACROSS THE 30 ° ANGLE MEASURES X. THE SIDE DIRECTLY ACROSS THE 60 ° ANGLE MEASURES X(SQRT 3). THE SIDE DIRECTLY ACROSS THE 90 ° ANGLE MEASURES 2X.
30°- 60°- 90° TRIANGLES
• • • 30° → X 60 °→ X(SQRT 3) 90 °→ 2X 30 ° X(SQRT 3) 2X 90 ° X 60 °
45 ° - 45 ° - 90 ° TRIANGLES
• • • THESE SPECIAL RIGHT TRIANGLES HAVE A 90 ° ANGLE, AND MUST HAVE TWO 45 ° ANGLES.
THE SIDES DIRECTLY ACROSS THE 45° ANGLES EACH MEASURE X. THE SIDE DIRECTLY ACROSS THE 90 ° ANGLE MEASURES X (SQRT 2).
45° - 45 ° - 90 ° TRIANGLES
• • • 45 °→ X 45 °→ X 90 ° → X(SQRT 2) 45° X X (SQRT 2) 90 ° X 45 °
REMINDER: ANGLES OF A TRIANGLE
• • the sum of the three angles of any triangle add up to 180 ° x ° + y °+z °= 180 ° x ° z ° y °
The drawing to the right shows 3 square parking lots that enclose a grassy area shaped like a right triangle.
If Lot A’s perimeter is 300 yards and lot B’s perimeter is 400 yards, what is the perimeter of Lot C?
A 500 yd B 700 yd C 1400 yd D 2000 yd
Obj 7 - TAKS 2004 9th [8.7(C)]
Problem #6
Look at the right triangle shown to the right. Which of the following could be the triangle’s dimensions?
A 12, 16.8, 18.2
B 5.4, 10.6, 16 C 1.2, 1.6, 2 D 8, 10, 12.5
Obj 7 - TAKS 2004 10th [8.7(C)]
Problem #9
Which figure best represents a triangle with sides a, b, and c in which the relationship a 2 + b 2 = c 2 is always true?
Obj 7 - TAKS 2003 8th [8.7(C)]
Problem #21
What is the area of the largest square in the diagram?
A. 5 units B. 9 units 2 2 C. 16 units D. 25 units 2 2 y
-8 -7 -6 -5 -4 -3 -2 -1 9 8 7 6 5 4 3 -1 -2 -3 -4 -5 2 1 0 1 2 3 4 5 6 7 8 -6 -7 -8 9
x
Obj 7 - TAKS 2003 9th {8.7(C)]
Problem #24
The sides of squares can be used to form triangles. The areas of the squares that form right triangles have a special relationship.
Obj 7 - TAKS 2003 10th [8.7(C)]
Problem #29
Using the dimensions of the squares shown below screen, determine which set of squares will form a right triangle.
Obj 7 - TAKS 2003 10th [8.7(C)]
Problem #29
A fence around a square garden has a perimeter of 48 feet. Find the approximate length of the diagonal of this square garden.
A. 12 feet B. 17 feet C. 21 feet D. 24 feet Problem #33 Obj 6 - TAKS 2003 11th [G.C1(C)]
Use the Pythagorean theorem to find the figure that is a right triangle.
A.
15 C.
15 12 20 27 20 B.
18 30 25 D.
25 60 65
Obj 7 - TAKS 2003 10th [8.7(C)]
Problem #31
A slide was installed at the local swimming pool, as shown below.
Which is closest to the length of the slide?
A. 29 ft B. 16 ft C. 21 ft D. 81 ft
Obj 8 - TAKS 2006 10th 8.9(A)]
Problem #53
Mr. Carpenter built a wooden gate, as shown below.
Which is closest to the length in feet of the diagonal board that Mr. Carpenter used to brace the wooden gate?
A. 4.9 ft B. 5.3 ft C. 6.1 ft D. 6.9 ft
Obj 8 - TAKS 2006 9th 8.9(A)]
Problem #46
The drawing shows part of the plan for a new underground lawn-sprinkler system.
Plan for Sprinkler System Plastic pipe 4 feet
A B C
4 feet Which is closest to the length of the section of plastic pipe from point A to point C? A. 4.7 ft B. 5.7 ft C. 6.7 ft D. 7.7 ft
Obj 8 - TAKS 2003 10th [8.9(A)]
Problem #32
The total area of trapezoid FGHJ is 52 square inches. What is the approximate length of segment FJ?
8 inches
G
A. 8.0 inches B. 8.5 inches C. 11.0 inches D. 11.5 inches
H
5 inches 8 inches
F J
Problem #41
Obj 8 - TAKS 2003 11th G.E1(C)]
In a town there is a small garden shaped like a triangle, as shown to the right. The side of the garden that faces Sixth Street is 80 feet in length. The side of the garden that faces Third Avenue is 30 feet in length. What is the approximate length of the side of the garden that faces Elm Street?
A. 35 ft B. 40 ft C. 85 ft D. 110 ft
Obj 8 - TAKS 2003 9th [8.9(A)]
Problem #31
Kim walked diagonally across a rectangular field that measured 100 feet by 240 feet.
Which expression could be used to determine how far Kim walked?
A
240
B
100 240
C
100 240 2
D
100 2 240 2 Obj 8 - TAKS 2004 11th [G.E1(C)]
Problem #24
Mrs. Cheung hired a landscaping service to plant a row of bushes around her triangular backyard.
If the bushes must be planted 3 feet apart, approximately how many bushes are needed for Mrs. Cheung’s backyard?
A 23 B 25 C 28 D 32
Obj 8 - TAKS 2004 10th [8.9(A)]
Problem #16
A square park has a diagonal walkway from 1 corner to another. If the walkway is about 38 yards long, what is the approximate length of each side of the park?
A 6 yd B 19 yd C 27 yd D 54 yd Problem #12
Obj 8 - TAKS 2004 9th [8.9(A)]
A cell-phone tower that has a transmission range of 50 miles is located 40 miles due south of a straight road. Find x, the length of the section of road that is within the transmission range of the tower.
A. 10 mi B. 30 mi C. 60 mi D. 90 mi
Obj 8 - TAKS 2003 8th [8.9(A)]
Problem #6
Mr. Elliott designed a flower garden in the shape of a square. He plans to build a walkway through the garden, as show to the right. Which is closest to the length of the walkway?
A 36 ft B 24 ft C 17 ft D 13 ft
Obj 8 - TAKS 2004 8th [8.9(A)]
Problem #1
On the map below, First Avenue and Second Avenue are parallel. A city planner proposes to locate a small garden and park on the triangular island by the intersections of four streets shown. What are the measures of the three angles of the garden?
A. 90 °, 65°, 25° B. 90 °, 50°, 40° C. 90 °, 60°, 30° D. 130 °, 40°, 10° Obj 6 - TAKS 2003 11th [G.C1(A)] Problem #32
Look at the drawing shown to the right.
If KMP is a right triangle formed by the placement of 3 squares, what is the area of the shaded square?
A. 135 in.
2 B. 24 in.
2 C. 66 in.
2 D. 81 in.
2
Obj 7 - TAKS 2006 9th [8.7(C)]
Problem #41
The lengths of the bases of an isosceles trapezoid are shown to the right. If the perimeter of this trapezoid is 32 units, what is its area?
A 44 square units B 110 square units C 88 square units D 55 square units Obj 6 - TAKS 2004 11th [G.C1(C)] Problem #15
The drawing below shows how 3 squares can be joined at their vertices to form a right triangle.
Which is closest to the area in square inches of the largest square?
A. 1914 in.
2 B. 233 in.
2 C. 210 in.
2 D. 1073 in.
2
Obj 7 - TAKS 2006 10th [8.7(C)]
Problem #46
The drawing to the right shows 3 square parking lots that enclose a grassy area shaped like a right triangle.
If Lot A’s perimeter is 300 yards and lot B’s perimeter is 400 yards, what is the perimeter of Lot C?
A 500 yd B 700 yd C 1400 yd D 2000 yd
Obj 7 - TAKS 2004 9th [8.7(C)]
Problem #6
Look at the right triangle shown to the right. Which of the following could be the triangle’s dimensions?
A 12, 16.8, 18.2
B 5.4, 10.6, 16 C 1.2, 1.6, 2 D 8, 10, 12.5
Obj 7 - TAKS 2004 10th [8.7(C)]
Problem #9
A wooden pole was broken during a windstorm. Before it broke, the total height of the pole above the ground was 25 feet. After it broke, the top of the pole touched the ground 15 feet from the base.
How tall was the part of the pole that was left standing?
A 8 ft B 10 ft C 17 ft D 20 ft
Obj 10 - TAKS 2004 11th [8.14(B)]
Problem #21
Mrs. Aman asked her students to look at the drawing shown below to determine the length of d.
Which of the following student responses best represents the length of d?
A. 8 units B. 11 units C. 14 units D. 3 units
Obj 10 - TAKS 2006 10th [8.15(A)]
Problem #89
Mr. Schultz has a garden shaped like an equilateral triangle that measures 11 feet on each side. He has placed a watering hose that extends from the faucet located at a vertex to the opposite side, as shown below.
Which is closest to the length of the hose in the garden?
A. 7.8 ft B. 9.5 ft C. 6.4 ft D. 5.5 ft Obj 6 - TAKS 2006 11th [G.C1(C)] Problem #49
WXY is isosceles.
WY is 10 centimeters long. Find the length of XZ.
F. 5 cm G. 10 cm H. 12 cm J. 13 cm
Obj 8 - TAKS 2006 11th G.E1(C)]
Problem #54
A kite string is 220 feet long from the kite to the ground. The string makes a 45 ° angle with the ground.
About how high off the ground is kite?
A 110 ft B 127 ft C 156 ft D 311 ft Obj 6 - TAKS 2004 11th [G.C1(C)] Problem #12
Mr. Ryan is flying his single-engine plane at an altitude of 2400 feet. He sees a cornfield at an angle of depression of 30 °.
What is Mr. Ryan’s approximate horizontal distance from the cornfield at this point?
A. 1200 feet B. 3394 feet C. 4157 feet D. 4800 feet Problem #36 Obj 6 - TAKS 2003 11th [G.C1(C)]
Look at the cube shown below.
Which equation best represents the area of the shaded rectangle located diagonally in the cube?
A.
A
= B.
A
= C. D.
A A
= = Problem #50 Obj 6 - TAKS 2006 11th [G.B4(A)]
In the figure shown below,
MN
= 10 centimeters.
Which is closest to the length of
TN
?
A. 7 cm B. 6 cm C. 17 cm D. 14 cm Obj 6 - TAKS 2006 11th [G.C1(C)] Problem #54