Question

You are building a set of nested tables. The surfaces of the tables will be $45{}^{\circ }\text{-}45{}^{\circ }\text{-}90°$triangles.

1. Each of the two congruent edges of the surface of the smallest table has a length of 24 inches. Make and label a scale drawing of the surface of the smallest table.
2. The ratio of an edge length of the surface of the smallest table to a corresponding edge length of the largest table is$1:2$. Find the length of each of the two congruent edges of the surface of the largest table.
3. The ratio of an edge length of the surface of the middle-sized table to a corresponding edge length of the surface of the smallest table is$3:2$. Find the length of each of the two congruent edges of the surface of the middle-sized table.
4. For each table surface, find the length of the third edge to the nearest inch. Make and label scale drawings of the surfaces of the two larger tables.

Short answer

1. $1:24$
2. 48 in
3. 36 in

(d) Smallest table:34 in

Middle size table:51 in

Largest table:68 in

Step-by-step solution

Step 1. Given Information.

A set of nested tables. The surface of the tables will be $45{}^{\circ }\text{-}45{}^{\circ }\text{-}90°$triangle.

Each of the two congruent edges of the surface of the smallest table has a length of 24 inches.

Step 2. Calculation.

First draw smallest triangle.

Let the scale of congruent side be$1:24$.

1 unit is equivalent to 24 in.

So, scale drawing would be as shown below.

Step 1. Given Information.

A set of nested tables. The surface of the tables will be $45{}^{\circ }\text{-}45{}^{\circ }\text{-}90°$triangle.

Each of the two congruent edges of the surface of the smallest table has a length of 24 inches.

Step 2. Calculation.

The length of congruent edges of the surface of the smallest table is 24 in.

The ratio of an edge length of the surface of the smallest table to a corresponding edge of the largest table is$1:2$.

Let the length of the surface of the largest table be x in.

So, the ratio is $24:x$ which is equal to$1:2$.

Therefore,

$\begin{array}{c}\frac{24}{x}=\frac{1}{2}\\ x=48\text{in}\end{array}$

Step 1. Given Information.

A set of nested tables. The surface of the tables will be $45{}^{\circ }\text{-}45{}^{\circ }\text{-}90°$triangle.

Each of the two congruent edges of the surface of the smallest table has a length of 24 inches.

Step 2. Calculation.

The length of congruent edges of the surface of the smallest table is 24 in.

The ratio of an edge length of the surface of the middle-sized table to smallest table is$3:2$.

Let the length of the surface of the middle-sized table be x in.

So, the ratio is $x:24$ which is equal to$3:2$.

Therefore,

$\begin{array}{c}\frac{x}{24}=\frac{3}{2}\\ x=36\text{in}\end{array}$

Step 1. Given Information.

A set of nested tables. The surface of the tables will be $45{}^{\circ }\text{-}45{}^{\circ }\text{-}90°$triangle.

Each of the two congruent edges of the surface of the smallest table has a length of 24 inches.

Step 2. Calculation.

The scale of each table.

So, the ratio of the congruent sides and third side is$1:1:\sqrt{2}$.

For smallest table, length of third side x,

$\begin{array}{c}\frac{1}{\sqrt{2}}=\frac{24}{x}\\ x\approx 34\text{in}\end{array}$

For middle sized table, length of third side y,

$\begin{array}{c}\frac{1}{\sqrt{2}}=\frac{36}{y}\\ y\approx 51\text{in}\end{array}$

For largest table, length of third side z,

$\begin{array}{c}\frac{1}{\sqrt{2}}=\frac{48}{z}\\ z\approx 68\text{in}\end{array}$