Question

18 × 32.4
16.2 × 18 ÷ 9 = 32.4

1. determine the length of bd in these similar triangles.

triangle diagram with labels: a, b, c, d, e; ae = 20, ed = 10.5, cd = 7
topics i should practice:

Explanation:

Step1: Identify similar triangles property

Since triangles \( \triangle CDE \) and \( \triangle CAE \) (or \( \triangle CDB \) and \( \triangle CAE \)) are similar, the ratios of corresponding sides are equal. Let \( BD = x \), then \( AE = 20 \), \( CD = 7 \), \( CE = 10.5 + 7 = 17.5 \)? Wait, no, wait. Wait, the sides: \( CE \) is \( 10.5 + 7 \)? Wait, no, looking at the diagram: \( E \) to \( D \) is \( 10.5 \), \( D \) to \( C \) is \( 7 \). So \( CE = ED + DC = 10.5 + 7 = 17.5 \)? Wait, no, maybe \( \triangle CBD \sim \triangle CAE \). So the ratio of \( CD \) to \( CE \) should equal the ratio of \( BD \) to \( AE \). Wait, let's correct: Let's denote \( \triangle CBD \sim \triangle CAE \) (since \( BD \parallel AE \), by basic proportionality theorem or similar triangles). So \( \frac{CD}{CE} = \frac{BD}{AE} \). Wait, \( CE = ED + DC = 10.5 + 7 = 17.5 \)? Wait, no, \( ED \) is \( 10.5 \), \( DC \) is \( 7 \), so \( CE = 10.5 + 7 = 17.5 \)? Wait, no, maybe \( CE \) is \( 10.5 \) and \( CD \) is \( 7 \)? Wait, the diagram: \( E \)---\( D \) (10.5)---\( C \), and \( A \)---\( B \)---\( C \), with \( AE = 20 \), \( BD \) is what we need. So triangles \( \triangle CBD \) and \( \triangle CAE \) are similar (AA similarity, since \( \angle C \) is common, and \( BD \parallel AE \), so \( \angle CBD = \angle CAE \), \( \angle CDB = \angle CEA \)). Therefore, the ratio of corresponding sides: \( \frac{CD}{CE} = \frac{BD}{AE} \). Wait, \( CE \) is \( ED + DC = 10.5 + 7 = 17.5 \)? Wait, no, \( ED \) is \( 10.5 \), \( DC \) is \( 7 \), so \( CE = 10.5 + 7 = 17.5 \). Then \( CD = 7 \), \( CE = 17.5 \), \( AE = 20 \), \( BD = x \). So \( \frac{7}{17.5} = \frac{x}{20} \). Wait, that can't be, because \( 7/17.5 = 0.4 \), so \( x = 20 * 0.4 = 8 \)? Wait, no, maybe I got the sides reversed. Wait, maybe \( \frac{CD}{CE} = \frac{BD}{AE} \), but \( CE = ED = 10.5 \)? No, that doesn't make sense. Wait, maybe \( CE = 10.5 \), \( CD = 7 \), so \( ED = CE - CD = 10.5 - 7 = 3.5 \)? No, the diagram shows \( E \) to \( D \) is \( 10.5 \), \( D \) to \( C \) is \( 7 \). So \( CE = ED + DC = 10.5 + 7 = 17.5 \). Then \( \frac{CD}{CE} = \frac{7}{17.5} = \frac{2}{5} \). Then \( BD = AE * \frac{CD}{CE} = 20 * \frac{7}{17.5} \). Wait, \( 17.5 = 35/2 \), so \( 7 / (35/2) = 7 * 2/35 = 14/35 = 2/5 \). Then \( 20 * 2/5 = 8 \). Wait, but let's check again. Alternatively, maybe the ratio is \( \frac{DC}{DE} = \frac{BD}{AE} \)? Wait, no, \( DE = 10.5 \), \( DC = 7 \), so \( \frac{DC}{DE} = 7/10.5 = 2/3 \). Then \( BD = AE * \frac{DC}{DE} = 20 * (7/10.5) \). Wait, \( 10.5 = 21/2 \), so \( 7 / (21/2) = 7 * 2/21 = 14/21 = 2/3 \). Then \( 20 * 2/3 \approx 13.33 \)? No, that's conflicting. Wait, maybe I mixed up the sides. Let's start over. Let's denote: \( \triangle CBD \sim \triangle CAE \). So \( \frac{CD}{CE} = \frac{BD}{AE} \). Wait, \( CE \) is the side from \( C \) to \( E \), which is \( CD + DE = 7 + 10.5 = 17.5 \). \( CD \) is 7, \( AE \) is 20, \( BD \) is x. So \( \frac{7}{17.5} = \frac{x}{20} \). Solving for x: \( x = 20 * (7 / 17.5) \). Calculate 7 / 17.5: 17.5 is 35/2, so 7 divided by 35/2 is 7 * 2/35 = 14/35 = 2/5. Then 20 * 2/5 = 8. So BD is 8. Wait, let's check: 7/17.5 = 0.4, 8/20 = 0.4. Yes, that works. So the ratio is 0.4, so BD = 8.

Wait, maybe the correct ratio is \( \frac{CD}{CE} = \frac{BD}{AE} \), where \( CE = ED + DC = 10.5 + 7 = 17.5 \), \( CD = 7 \), \( AE = 20 \). So \( \frac{7}{17.5} = \frac{BD}{20} \). Solving for BD: BD = 20 * (7 / 17.5) = 20 * 0.4 = 8.

Step1: Set up proportion for similar triangles

Since \( \triangle CBD \sim \triangle CAE \) (AA similarity), the ratios of corresponding sides are equal:
\( \frac{CD}{CE} = \frac{BD}{AE} \)

Step2: Identify side lengths

- \( CD = 7 \)
- \( CE = ED + CD = 10.5 + 7 = 17.5 \)
- \( AE = 20 \)
- Let \( BD = x \) (unknown length)

Step3: Substitute into proportion

Substitute the known values into the proportion:
\( \frac{7}{17.5} = \frac{x}{20} \)

Step4: Solve for \( x \) (BD)

Cross - multiply to solve for \( x \):
\( 17.5x = 7 \times 20 \)
\( 17.5x = 140 \)
Divide both sides by \( 17.5 \):
\( x=\frac{140}{17.5}=8 \)

Answer:

The length of \( BD \) is \( \boldsymbol{8} \).