a To find the area of the house, we will add a segment to the diagram creating a 30^(∘)-60^(∘)-90^(∘) triangle with legs that we will label x and y.
By determining x we can find the length of the four congruent sides at the top of the diagram. In a 30^(∘)-60^(∘)-90^(∘) triangle, the shorter leg is half the length of the hypotenuse and the longer leg is sqrt(3) times longer than the shorter leg. With this information we can find the length of the legs.
If the leg labeled x is 15 mm, the side that is (51-x) mm must be 36 mm. We can determine that the length of the congruent sides is 363=12 mm. In addition to the triangle, we will divide the rest of the shape into rectangles.
Now we can calculate the total area of the shape.
2(15sqrt(3)(12))+(15sqrt(3)-12)12+(15)(15sqrt(3))/2
⇕
986.16 mm^2
The area is about 986.16 mm^2.
b After the enlargement the new diagram is 400 %, or 4 times, larger than that of the original. This means the linear scale factor is 4. To find the ratio of the areas we have to square the linear scale factor, which will give us the area scale factor.
(Linear scale factor)^2=( 4)^2
⇕
Area scale factor=16
The area of the new diagram is 16 times greater than that of the original diagram. From Part A, we know that the original diagram had an area of 986.16 mm^2. The area of the new diagram is 16 times this.
16(986.16)≈ 15 779 mm^2
