Question

9)use the diagram to solve for x, how high up the ladder will go?

Explanation:

Step1: Find the height of the smaller right - triangle

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) for the smaller right - triangle with hypotenuse \(24\). Let the vertical side be \(y\), then \(y=\sqrt{24^{2}-10^{2}}=\sqrt{576 - 100}=\sqrt{476}\approx21.8\).

Step2: Find the height of the larger right - triangle

For the larger right - triangle with hypotenuse \(30\), let the base be \(b\). Since the two right - triangles are similar (by AA similarity, as the angles are equal), or we can use the property of parallel lines and right - angles.
First, find the base of the larger right - triangle. Let's assume the base of the larger right - triangle is \(b\).
We know that the two right - triangles are related.
We use the Pythagorean theorem for the larger right - triangle: \(x=\sqrt{30^{2}-b^{2}}\).
Another way:
We can use the fact that the two right - triangles are similar. But a more straightforward way is to use the Pythagorean theorem for the larger right - triangle.
Let's assume the base of the larger right - triangle is \(b\). We know from the smaller right - triangle, if we consider the proportion (not necessary here as we can use the Pythagorean theorem directly for the larger triangle).
\(x=\sqrt{30^{2}-24^{2}}=\sqrt{900 - 576}=\sqrt{324}=18\).

Answer:

\(18\)