Question

what is the value of s? units

Explanation:

Step1: Apply the Pythagorean theorem to \(\triangle ABD\)

In \(\triangle ABD\), if we assume the right - angled side \(AB = 8\), and the hypotenuse \(AD=s\). But we need to check the relationship between the triangles. Since \(\triangle CBD\) and \(\triangle ABD\) are right - angled triangles and we can use the geometric mean theorem (or the similarity of right - angled triangles).

We know that if we have two right - angled triangles \(\triangle ABC\) (right - angled at \(B\)) and \(\triangle ABD\) (right - angled at \(B\)) and \(\triangle CBD\) (right - angled at \(B\)), we can use the formula \(AD=\sqrt{AB^{2}+BD^{2}}\). But another way is using the similarity of \(\triangle ABC\) and \(\triangle ABD\).

Since \(\triangle ABC\sim\triangle ABD\) (by AA similarity, as \(\angle ABC=\angle ABD = 90^{\circ}\) and \(\angle A\) is common). Also, we can use the formula \(AD=\frac{AB\times(AB + BC)}{AB / \sqrt{AD^{2}-AB^{2}}}\). But a more straightforward way is using the formula for the length of the hypotenuse of a right - angled triangle formed by similar right - angled triangles.

We know that if we consider the two right - angled triangles \(\triangle ABC\) (with \(AB = 8\), \(BC = 5\)) and \(\triangle ABD\) (with \(BD=15\)). Using the formula \(s=\sqrt{(8 + 5)\times8+15^{2}}\) is wrong. The correct formula is based on the similarity of right - angled triangles.

We know that if we have two right - angled triangles \(\triangle ABC\) and \(\triangle ABD\) (right - angled at \(B\)), and we use the fact that \(\frac{AB}{BD}=\frac{BC}{AB}\) (from \(\triangle ABC\sim\triangle ABD\)) is wrong. The correct approach is using the Pythagorean theorem in \(\triangle ABD\) where \(AB = 8\), \(BD = 15\)

By the Pythagorean theorem \(s=\sqrt{8^{2}+15^{2}}\)

Step2: Calculate \(s\)

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Answer:

\(17\)