Apply Pythagoras' theorem in two-dimensional contexts

The supports are vertical and the ground is level. What is the vertical distance between their tops? The diagram is not to scale

A wire joins the tops of two vertical supports on level ground. The left support stays 9 m high. Lower the right support, then raise it through the full range. Watch the height difference and wire length. The diagram is not to scale

As the support rose through the full range, how did the wire length change?

Find the triangle before calculating

The heights of two objects are not usually the sides of the required triangle. Use the separation between the endpoints. You already know how to find a side once the right triangle is identified

1. 1Locate the right angle Find the two perpendicular segments. Vertical supports and level ground make a right angle
2. 2Find the relevant separation For tops measured from the same level ground, subtract the shorter height from the taller height
3. 3Choose the side direction Add squared legs for a hypotenuse. For an unknown leg, subtract the known leg's square from the hypotenuse's square \(\displaystyle \begin{aligned}c^2&=a^2+b^2\\a^2&=c^2-b^2\end{aligned}\)
4. 4Finish in the requested units Take the positive square root. Keep the calculator's full value until the final rounding

A taut line joins the tops of the vertical supports on level ground. Calculate its length, x, in metres, correct to one decimal place. The diagram is not to scale

Which part of the taller support lies above the shorter support? Give the final length to exactly one decimal place

A taut line joins the tops of the vertical supports on level ground. Calculate the distance between their bases, x, in metres. The diagram is not to scale

Find the height difference first. Which known length is opposite the right angle?

Use a shared side in two triangles

A flat frame can contain two right triangles with a shared side. The shared side can be the hypotenuse of the first triangle and a leg of the second

1. 1Start with enough information Use the triangle with two known sides to find the square of the shared side
2. 2Check the next right angle Identify the hypotenuse of the second triangle before deciding whether to add or subtract
3. 3Keep the square exact Carry the exact squared length into the second equation. There is no need to round the shared side

Calculate the frame's length AD in metres, correct to one decimal place. Both triangles lie in one plane. The diagram is not to scale

Start at the right angle at B. Keep the square of the shared side for the triangle with its right angle at C. Give the final length to exactly one decimal place

Taller support

\(\displaystyle 15\text{ m}\)

Shorter support

\(\displaystyle 8\text{ m}\)

Horizontal separation

\(\displaystyle 10\text{ m}\)

A line joins the tops of two vertical supports on level ground. The student is finding its length, d, using h for the vertical separation. Which line is the first error?