At the beginning of this lesson, you read about Olivia, who used the right triangle set of Buildz to create many different structures and then used triangle properties to calculate rectangular dimensions. You, too, can create design solutions using your knowledge of triangles, but first you need to know what types of problems you can solve using the relationship between triangles and rectangles. Study the problems and solutions presented below.
Problem 1
Problem 2
Problem 3
Olivia used 45-45 and 30-60 right triangle Buildz to design a house.
How tall is the base of the house (the square part without the roof)? Click each step below to see how to solve this problem.
| Step 1 | The base of the house contains several 30-60 right triangles. |
| Step 2 | Therefore, the side ratio is 1:2:\(\mathsf{\small\sqrt{3}}\). |
| Step 3 | The hypotenuse of △DEF is 6 m. |
| Step 4 | Because side \(\mathsf{ \small\overline{DE} }\) is opposite ∠60°, its length is 3\(\mathsf{\small\sqrt{3}}\). |
| Step 5 | Therefore, the base of the house is 3\(\mathsf{\small\sqrt{3}}\) m tall. |
Olivia used 45-45 and 30-60 right triangle Buildz to design a see-saw.
How long is the seat of the see-saw? Click each step below to see how to solve this problem.
| Step 1 | The seat of the see-saw contains two 30-60 right triangles. |
| Step 2 | Therefore, the side ratio is 1:2:\(\mathsf{\small\sqrt{3}}\). |
| Step 3 | The shortest side of △GJK is 12 cm. |
| Step 4 | Because side GK is opposite ∠60°, its length is 12\(\mathsf{\small\sqrt{3}}\) cm. |
| Step 5 | Therefore, the seat of the see-saw is 12\(\mathsf{\small\sqrt{3}}\) cm long. |
Now try finding the dimensions of a rectangle that does not contain special right triangles. In the figure below, ∠XWY has a tangent ratio of 10:4.
How long are the diagonals of rectangle WXYZ? Click each step below to see how to solve this problem.
| Step 1 | The tangent ratio is defined as opposite/adjacent. |
| Step 2 | Therefore, XY = 10 and XW = 4 |
| Step 3 | In rectangle WXYZ, the diagonal WY = \(\mathsf{\small\sqrt{{XY}^2 + {XW}^2}}\) |
| Step 4 | Therefore, WY = \(\mathsf{ \small\sqrt{10^2 + 4^2}}\) = 10.77 |
