Question

level 2 solve for the value of y

Explanation:

Step1: Apply the geometric mean theorem (altitude-on-hypotenuse theorem)

For a right - triangle with altitude \(h\) to the hypotenuse \(c\), and the two segments of the hypotenuse \(a\) and \(b\), the formula for the length of one of the non - hypotenuse sides \(y\) (where \(y\) is adjacent to segment \(a\) of the hypotenuse) is \(y=\sqrt{a\times c}\). Here, \(a = 13\) and \(c\) (the hypotenuse of the large right - triangle) is found using the Pythagorean theorem for the two right - triangles formed by the altitude. But using the geometric mean theorem directly: \(y^{2}=13\times(13)\) (incorrect approach, let's use the correct formula \(y^{2}=13\times(13)\) is wrong. The correct formula: In a right - triangle, if the hypotenuse is \(c = 13\) (wait no, the hypotenuse of the large triangle, assume the two segments of the hypotenuse from the altitude are \(x\) and \(13 - x\). But using the formula \(y^{2}=13\times(13)\) is wrong. Wait, the correct formula for the leg of a right - triangle in terms of the hypotenuse and the adjacent segment: If we have a right - triangle with hypotenuse \(c\) and an altitude \(h\) to the hypotenuse, dividing the hypotenuse into two segments \(m\) and \(n\). The formula for a leg \(y\) (adjacent to \(n\)) is \(y=\sqrt{n\times c}\). Here, assume the hypotenuse of the large right - triangle is \(13\) (no, wait the figure shows a right - triangle with a leg \(12\), hypotenuse of a sub - triangle. Wait, using the formula \(y^{2}=13\times(13)\) is wrong. Let's use the correct formula: In a right - triangle, if the hypotenuse is \(c\) and we have an altitude \(h\) to the hypotenuse, dividing it into two segments \(m\) and \(n\). The formula for a leg \(y\) (adjacent to \(n\)) is \(y = \sqrt{n\times c}\). Wait, another approach: Using the Pythagorean theorem. Let the two segments of the hypotenuse (from the altitude) be \(x\) and \(13 - x\). For the two smaller right - triangles: \(12^{2}=x\times13\) (by the geometric mean theorem \(h^{2}=m\times n\), where \(h\) is the altitude. Then \(x=\frac{144}{13}\). And \(y^{2}=(13)\times(13 - x)\). Substitute \(x=\frac{144}{13}\) into \(y^{2}=13\times(13-\frac{144}{13})\). \(y^{2}=13\times\frac{169 - 144}{13}\).

Step2: Simplify the expression

\(y^{2}=169 - 144\).
\(y^{2}=25\).
Take the square root of both sides: \(y = 5\).

Answer:

\(y = 5\)