Question

arielle is building the wooden framework for the roof of a house. she needs the angle created by the vertical and horizontal boards of the frame to be a right angle. the height of the vertical board is 12 feet. the length of the horizontal board is 15 feet. the support beam that will connect the ends of the two boards measures 20 feet. which is true regarding the triangular frame? it is an acute triangle. about 0.8 foot needs to be removed from the 20 - foot board to create a right triangle. it is an acute triangle. about 7 feet need to be removed from the 20 - foot board to create a right triangle. it is an obtuse triangle. about 0.8 foot needs to be removed from the 20 - foot board to create a right triangle. it is an obtuse triangle. about 7 feet need to be removed from the 20 - foot board to create a right triangle.

Explanation:

Step1: Apply Pythagorean theorem

Let the horizontal board be \(a = 15\) feet, the vertical board be \(b = 12\) feet. In a right - triangle, by the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(c\) is the hypotenuse of the right - triangle. So \(c=\sqrt{15^{2}+12^{2}}=\sqrt{225 + 144}=\sqrt{369}\approx19.2\) feet.

Step2: Find the length to be removed from 20 - foot board

The length of the 20 - foot board that needs to be removed to form a right - triangle is \(20-\sqrt{15^{2}+12^{2}}=20 - 19.2 = 0.8\) feet.

Step3: Determine the type of triangle

We can also use the cosine law to check the type of triangle with sides \(a = 15\), \(b = 12\) and \(c = 20\). The cosine law is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Substituting the values: \(20^{2}=15^{2}+12^{2}-2\times15\times12\times\cos C\), \(400=225 + 144-360\cos C\), \(400 = 369-360\cos C\), \(\cos C=\frac{369 - 400}{360}=\frac{- 31}{360}<0\). Since \(\cos C<0\), angle \(C\) is obtuse.

Answer:

It is an obtuse triangle. About 0.8 foot needs to be removed from the 20 - foot board to create a right triangle.