Solve the geometry construction problems from practice set 4.1

Steps to Solve

Here's how to approach these construction problems, broken down into individual steps for each triangle pair:

1. ΔABC ~ ΔLMN where BC/MN = 5/4

1. Construct ΔABC: Start by drawing the triangle $ABC$ with the given side lengths $AB = 5.5 \text{ cm}$, $BC = 6 \text{ cm}$, and $CA = 4.5 \text{ cm}$. You can do this using a ruler and compass.

2. Determine the side length of MN: We are given that $\frac{BC}{MN} = \frac{5}{4}$. We also know that $BC = 6 \text{ cm}$. We can use that to find $MN$. So, $$ \frac{6}{MN} = \frac{5}{4} $$ $$ MN = \frac{6 \times 4}{5} = \frac{24}{5} = 4.8 \text{ cm} $$

3. Find the scale factor: The scale factor $k$ from $\triangle ABC$ to $\triangle LMN$ is $\frac{4}{5} = 0.8$. Multiply each side length of $\triangle ABC$ by $0.8$ to find corresponding lengths of $\triangle LMN$: $LM = AB \times 0.8 = 5.5 \times 0.8 = 4.4 \text{ cm}$ $MN = BC \times 0.8 = 6 \times 0.8 = 4.8 \text{ cm}$ $LN = CA \times 0.8 = 4.5 \times 0.8 = 3.6 \text{ cm}$

4. Construct ΔLMN: Now that you have calculated all the sides of $\triangle LMN$, construct the triangle with the following side lengths: $LM = 4.4 \text{ cm}$, $MN = 4.8 \text{ cm}$, $LN = 3.6 \text{ cm}$.

2. ΔPQR ∼ ΔLTR where PQ/LT = 3/4

1. Construct ΔPQR: Draw $\triangle PQR$ with $PQ = 4.2 \text{ cm}$, $QR = 5.4 \text{ cm}$, and $PR = 4.8 \text{ cm}$.

2. Find the scale factor: The scale factor $k$ from $\triangle PQR$ to $\triangle LTR$ is $\frac{4}{3}$. Multiply each side length of $\triangle PQR$ by $\frac{4}{3}$ to find corresponding lengths of $\triangle LTR$: $LT = PQ \times \frac{4}{3} = 4.2 \times \frac{4}{3} = 5.6 \text{ cm}$ $TR = QR \times \frac{4}{3} = 5.4 \times \frac{4}{3} = 7.2 \text{ cm}$ $LR = PR \times \frac{4}{3} = 4.8 \times \frac{4}{3} = 6.4 \text{ cm}$

3. Construct ΔLTR: Construct $\triangle LTR$ with $LT = 5.6 \text{ cm}$, $TR = 7.2 \text{ cm}$, and $LR = 6.4 \text{ cm}$.

3. ΔRST ∼ ΔXYZ where RS/XY = 3/5

1. Construct ΔRST: Draw $RS = 4.5 \text{ cm}$. At point $S$, construct an angle $\angle RST = 40^\circ$. Make $ST = 5.7 \text{ cm}$. Connect $R$ and $T$ to form $\triangle RST$.

2. Find the scale factor: The scale factor $k$ from $\triangle RST$ to $\triangle XYZ$ is $\frac{5}{3}$. Multiply each side length of $\triangle RST$ by $\frac{5}{3}$ to find corresponding lengths of $\triangle XYZ$: $XY = RS \times \frac{5}{3} = 4.5 \times \frac{5}{3} = 7.5 \text{ cm}$ $YZ = ST \times \frac{5}{3} = 5.7 \times \frac{5}{3} = 9.5 \text{ cm}$

3. Determine $\angle XYZ$: Since $\triangle RST \sim \triangle XYZ$, $\angle XYZ = \angle RST = 40^\circ$.

4. Construct ΔXYZ: Draw $XY = 7.5 \text{ cm}$. At point $Y$, construct an angle $\angle XYZ = 40^\circ$. Make $YZ = 9.5 \text{ cm}$. Finally, connect $X$ and $Z$.

4. ΔAMT ∼ ΔAHE where AM/AH = 7/5

1. Construct ΔAMT: Draw $AM = 6.3 \text{ cm}$. At point $A$, create an angle $\angle TAM = 50^\circ$. Along this line, make $AT = 5.6 \text{ cm}$. Connect $M$ and $T$.

2. Find the length of AH: We have $\frac{AM}{AH} = \frac{7}{5}$. Since $AM = 6.3 \text{ cm}$, $$ \frac{6.3}{AH} = \frac{7}{5} $$ $$ AH = \frac{6.3 \times 5}{7} = 4.5 \text{ cm} $$

3. Find the length of AE: Find the scale factor $k$ from $\triangle AMT$ to $\triangle AHE$ is $\frac{5}{7}$. Multiply the side length of $\triangle AT$ by $\frac{5}{7}$ to find the side length of $\triangle AE$. $AE = AT \times \frac{5}{7} = 5.6 \times \frac{5}{7} = 4 \text{ cm}$

4. Determine $\angle HAE$: Since $\triangle AMT \sim \triangle AHE$, the angle $\angle HAE = \angle TAM = 50^\circ$.

5. Construct ΔAHE: Draw $AH = 4.5 \text{ cm}$. At point $A$, create an angle $\angle HAE = 50^\circ$. Along this line, make $AE = 4 \text{ cm}$. Connect $H$ and $E$.