Question

this is a diagram of the roof of a house, when viewed from the side. if the house is $14\sqrt{2}$ feet wide, how long is one of the rooftop slanted sides? 14\sqrt{2} feet 45° 45°

Explanation:

Step1: Identify the triangle type

The triangle formed by the roof has two angles of \(45^\circ\), so the third angle is \(180^\circ - 45^\circ - 45^\circ = 90^\circ\). It's an isosceles right triangle. The base of the triangle (the width of the house part under the roof) is \(14\sqrt{2}\) feet. In an isosceles right triangle, the legs are equal, and the hypotenuse \(c\) is related to the leg \(a\) by \(c = a\sqrt{2}\), but here we can also use trigonometry or the properties of 45 - 45 - 90 triangles. Wait, actually, the base of the triangle (the horizontal side) is \(14\sqrt{2}\), and we can split the isosceles triangle into two right triangles by drawing a perpendicular from the top vertex to the base. This perpendicular bisects the base, so each right triangle has a base of \(\frac{14\sqrt{2}}{2}=7\sqrt{2}\) feet and an angle of \(45^\circ\) at the corner of the roof.

Step2: Use cosine to find the slanted side

Let the slanted side be \(x\). In the right triangle (half of the isosceles triangle), \(\cos(45^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7\sqrt{2}}{x}\). We know that \(\cos(45^\circ)=\frac{\sqrt{2}}{2}\), so \(\frac{\sqrt{2}}{2}=\frac{7\sqrt{2}}{x}\). Cross - multiplying gives \(x\times\sqrt{2}=2\times7\sqrt{2}\). Dividing both sides by \(\sqrt{2}\), we get \(x = 14\) feet? Wait, no, wait. Wait, maybe I made a mistake. Wait, the base of the isosceles triangle is \(14\sqrt{2}\), and the slanted side is a leg of the right triangle? No, wait, no. Wait, the slanted side is the hypotenuse of the right triangle we formed? Wait, no, let's re - examine. The roof's slanted side is a side of the isosceles triangle. The isosceles triangle has two equal sides (the slanted sides) and a base of \(14\sqrt{2}\). The base angles are \(45^\circ\), so using the law of sines: \(\frac{\text{side opposite }45^\circ}{\sin(45^\circ)}=\frac{\text{side opposite }90^\circ}{\sin(90^\circ)}\). The side opposite \(90^\circ\) is the base \(14\sqrt{2}\), and the side opposite \(45^\circ\) is the slanted side \(x\). So \(\frac{x}{\sin(45^\circ)}=\frac{14\sqrt{2}}{\sin(90^\circ)}\). Since \(\sin(45^\circ)=\frac{\sqrt{2}}{2}\) and \(\sin(90^\circ)=1\), we have \(x=\frac{14\sqrt{2}\times\sin(45^\circ)}{1}=\frac{14\sqrt{2}\times\frac{\sqrt{2}}{2}}{1}\). Simplify \(\sqrt{2}\times\sqrt{2}=2\), so \(x = \frac{14\times2}{2}=14\)? Wait, no, that can't be. Wait, maybe the base of the isosceles triangle is not \(14\sqrt{2}\), but the width of the house is \(14\sqrt{2}\), which is the length of the rectangle's top side, and the isosceles triangle is on top of the rectangle. So the base of the isosceles triangle is equal to the width of the house, which is \(14\sqrt{2}\). In an isosceles triangle with two \(45^\circ\) angles, the sides (the slanted sides) can be found by using the formula for the side of an isosceles triangle. Let's use the law of cosines: \(x^{2}=x^{2}+(14\sqrt{2})^{2}-2\times x\times14\sqrt{2}\times\cos(45^\circ)\). Wait, no, that's for two sides and the included angle. Wait, no, the isosceles triangle has two sides \(x\) and base \(b = 14\sqrt{2}\), and the included angle between the two sides \(x\) is \(90^\circ\)? No, earlier we thought the third angle is \(90^\circ\), so it's an isosceles right triangle, so the two equal sides are the legs, and the base is the hypotenuse. Wait, that's the mistake! If the triangle has angles \(45^\circ\), \(45^\circ\), \(90^\circ\), then the two equal sides are the legs (length \(a\)), and the hypotenuse (the base of the isosceles triangle) is \(a\sqrt{2}\). So if the hypotenuse (the base of the isosceles triangle) is \(14\sqrt{2}\), then \(a\sqrt{2}=14\sqrt{2}\), so \(a = 14\). Wait, the legs are the slanted sides? No, no. Wait, in a 45 - 45 - 90 triangle, the legs are the two equal sides, and the hypotenuse is the side opposite the right angle. So if the base of the isosceles triangle (the side opposite the right angle) is \(14\sqrt{2}\), then the legs (the slanted sides of the roof) are equal to the length of the legs of the right triangle, which are \(a\), and \(a\sqrt{2}=14\sqrt{2}\), so \(a = 14\). Wait, that makes sense. So the slanted side (the leg of the 45 - 45 - 90 triangle) is 14 feet? Wait, no, let's draw it mentally. The roof is an isosceles triangle on top of a rectangle. The width of the house is the length of the rectangle's top side, which is equal to the base of the isosceles triangle. The isosceles triangle has two angles of \(45^\circ\) at the base (the rectangle's top side), so the vertex angle is \(90^\circ\). So it's an isosceles right triangle, where the base (the side along the rectangle) is the hypotenuse of the isosceles right triangle, and the two equal sides are the slanted sides of the roof. Wait, that's the key mistake. So if the hypotenuse (base of the triangle) is \(14\sqrt{2}\), and in a 45 - 45 - 90 triangle, hypotenuse \(= \text{leg}\times\sqrt{2}\), so leg \(=\frac{\text{hypotenuse}}{\sqrt{2}}\). So leg \(=\frac{14\sqrt{2}}{\sqrt{2}} = 14\). So the slanted side (the leg of the 45 - 45 - 90 triangle) is 14 feet? Wait, no, the slanted side is the leg, and the base of the triangle is the hypotenuse. So yes, if the hypotenuse is \(14\sqrt{2}\), then the leg (slanted side) is \(\frac{14\sqrt{2}}{\sqrt{2}}=14\) feet.

Wait, another way: use sine. In the right triangle (half of the isosceles triangle), the angle at the roof corner is \(45^\circ\), the opposite side to the \(45^\circ\) angle is the height of the triangle, and the hypotenuse is the slanted side. But we can also use the fact that in the isosceles triangle with base \(14\sqrt{2}\) and two \(45^\circ\) angles, the sides (slanted sides) can be found by the formula for the side of a triangle: \(\frac{x}{\sin(45^\circ)}=\frac{14\sqrt{2}}{\sin(90^\circ)}\). Since \(\sin(90^\circ)=1\) and \(\sin(45^\circ)=\frac{\sqrt{2}}{2}\), then \(x=\frac{14\sqrt{2}\times\sin(45^\circ)}{1}=\frac{14\sqrt{2}\times\frac{\sqrt{2}}{2}}{1}=\frac{14\times2}{2}=14\). So the length of one slanted side is 14 feet? Wait, no, that seems too short. Wait, maybe the width of the house is the length of the rectangle, and the base of the triangle is equal to the width of the house. Wait, maybe I mixed up the triangle. Let's consider that the roof's slanted side is a side of a right triangle where the adjacent side to the \(45^\circ\) angle is half of the house's width. The house's width is \(14\sqrt{2}\), so half of it is \(7\sqrt{2}\). Then, \(\cos(45^\circ)=\frac{7\sqrt{2}}{x}\), so \(x=\frac{7\sqrt{2}}{\cos(45^\circ)}\). Since \(\cos(45^\circ)=\frac{\sqrt{2}}{2}\), then \(x=\frac{7\sqrt{2}}{\frac{\sqrt{2}}{2}}=7\sqrt{2}\times\frac{2}{\sqrt{2}} = 14\). Oh, so that's correct. So the slanted side is 14 feet.

Answer:

The length of one of the rooftop slanted sides is \(\boldsymbol{14}\) feet.