Question 85·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

0 / 200 Questions completed

A vertical flagpole is perpendicular to level ground. A wire of length $40$ feet extends from the top of the flagpole to a ground anchor on one side of the flagpole's base. A second wire of length $74$ feet extends from the top to a ground anchor on the opposite side of the base. The two anchors and the flagpole's base are collinear, with the base between the anchors, and the anchors are $102$ feet apart.

What is the height, in feet, of the flagpole?

Your answer

A pole perpendicular to level ground forms two right triangles, so use the Pythagorean theorem: the legs’ squares add to the slanted side’s square. Anchors on opposite sides have ground distances that add to their separation. The triangles share one height, so their expressions for its square must match. Use a Desmos regression to find a ground distance, then calculate the height.

Hints

Hint 1
A leg of a right triangle is one of the two sides that make the right angle. Call the ground distance to the shorter wire’s anchor $a$ . Since the base lies between the anchors, how would you express the other ground distance?
Hint 2
The hypotenuse is the side opposite a right angle, so each wire is a hypotenuse. Use the Pythagorean theorem with each wire separately; the entire distance between anchors is not a leg of either triangle.
Hint 3
Both triangles share the same pole height. Write an expression for its square using each wire, then set those expressions equal. Once you find a ground distance, which triangle can you use to find the height?

Step-by-step

Match the two right triangles
Step 1
Find an expression for the other ground distance
Let $a$ be the ground distance from the base to the $40$ -foot wire’s anchor, and $b$ the distance to the other anchor. The base is between the anchors, so their distances add to $102$ feet:
$a+b=102.$
Subtract $a$ to express the other distance:
$b=102-a.$
Step 2
Use the shorter wire to express the height squared
Let $h$ be the pole’s height. The pole is perpendicular to the ground, so the $40$ -foot wire is the hypotenuse, the side opposite the right angle. The Pythagorean theorem says the legs’ squares add to the hypotenuse’s square:
$h^2+a^2=40^2.$
Subtract $a^2$ :
$h^2=40^2-a^2.$
Step 3
Use the longer wire to express the same height squared
The $74$ -foot wire is the other hypotenuse. Its ground leg is $102-a$ , not the entire distance between anchors. Apply the same theorem:
$h^2+(102-a)^2=74^2.$
Subtract $(102-a)^2$ :
$h^2=74^2-(102-a)^2.$
Step 4
Set the height expressions equal
There is one pole height, so both expressions for $h^2$ must have the same value. Set them equal:
$40^2-a^2=74^2-(102-a)^2.$
Step 5
Solve for the first ground distance
Type $40^2-x_{1}^2\sim74^2-(102-x_{1})^2$ in Desmos. Use $x_{1}$ in place of $a$ so Desmos solves for that distance, and use $\sim$ to tell it to make the two sides match. Under PARAMETERS, Desmos shows $x_{1}=32$ . So the first anchor is $32$ feet from the base; that is not yet the height.
Step 6
Calculate the pole’s height
From the shorter wire’s triangle, $h^2=40^2-a^2$ . A length is positive, so type $\sqrt{40^2-x_{1}^2}$ beneath the regression. Desmos displays $24$ . The flagpole is $24$ feet tall. Grid in 24.

Question 85·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this