Question

1. the triangles are shown

what are the possible values of g and h?
a) g = 4, h = 7
b) g = 6, h = 8
c) g = 10, h = 13
d) g = 9, h = 18

Explanation:

Step1: Identify Similar Triangles

The triangles are similar (same shape, parallel sides imply similar by AA similarity). So, the ratios of corresponding sides should be equal. The small triangle has base 3 and height 6; the large triangle has base \( g \) and height \( h \). So, \(\frac{3}{g}=\frac{6}{h}\) or simplified \(\frac{1}{2}=\frac{3}{g}=\frac{6}{h}\) (wait, no: \(\frac{3}{g}=\frac{6}{h}\) simplifies to \( h = 2g \)? Wait, no, small triangle: base 3, height 6. Large triangle: base \( g \), height \( h \). So ratio of base to height for small: \( \frac{3}{6}=\frac{1}{2} \). So for large, \( \frac{g}{h}=\frac{1}{2} \) → \( h = 2g \)? Wait, no, maybe I mixed up. Wait, the small triangle's vertical side is 6, horizontal is 3. The large triangle's vertical is \( h \), horizontal is \( g \). Since they are similar, the ratio of corresponding sides should be equal. So \( \frac{3}{g}=\frac{6}{h} \) → cross-multiplying: \( 3h = 6g \) → \( h = 2g \). Wait, but let's check the options:

a) \( g=4, h=7 \): \( 7 = 2*4? 8≠7 \) → no.

b) \( g=6, h=8 \): \( 8=2*6? 12≠8 \) → no.

c) \( g=10, h=13 \): \( 13=2*10? 20≠13 \) → no. Wait, maybe I got the ratio reversed. Maybe \( \frac{6}{h}=\frac{3}{g} \) → \( \frac{6}{3}=\frac{h}{g} \) → \( 2=\frac{h}{g} \) → \( h = 2g \). Wait, d) \( g=9, h=18 \): \( 18=2*9 \) → yes! Wait, but wait, maybe the small triangle is base 3, height 6; large is base \( g \), height \( h \). So the ratio of small to large should be consistent. Wait, maybe I made a mistake. Wait, let's check the ratio of the sides. The small triangle: horizontal 3, vertical 6. So the ratio of horizontal to vertical is 3:6 = 1:2. So the large triangle should have horizontal to vertical ratio 1:2, so \( g:h = 1:2 \) → \( h = 2g \). Now check options:

a) \( g=4 \), \( h=7 \): 7 vs 8 (2*4=8) → no.

b) \( g=6 \), \( h=8 \): 8 vs 12 (2*6=12) → no.

c) \( g=10 \), \( h=13 \): 13 vs 20 (2*10=20) → no.

d) \( g=9 \), \( h=18 \): 18=29 → yes! Wait, but wait, maybe the ratio is vertical to horizontal: 6:3=2:1, so \( h:g=2:1 \) → \( h=2g \). So d) is \( h=18 \), \( g=9 \), so 18=29. That works. Wait, but let's check the other way. Maybe the small triangle's horizontal is 3, vertical is 6; large horizontal is \( g \), vertical is \( h \). So the scale factor: if we take the small to large, what's the scale? Let's see the options. Let's check the ratio of \( g \) to 3 and \( h \) to 6. So \( \frac{g}{3}=\frac{h}{6} \) → \( h = 2g \). So for d) \( g=9 \), \( h=18 \): \( 9/3=3 \), \( 18/6=3 \). Oh! Wait, that's the scale factor. So scale factor is 3. So small triangle: 3 (base), 6 (height). Large: 33=9 (base), 63=18 (height). So that's a scale factor of 3. So that works. Let's check other options:

a) \( g=4 \): 4/3 ≈1.333, \( h=7 \):7/6≈1.166 → not same scale.

b) \( g=6 \):6/3=2, \( h=8 \):8/6≈1.333 → not same.

c) \( g=10 \):10/3≈3.333, \( h=13 \):13/6≈2.166 → not same.

d) \( g=9 \):9/3=3, \( h=18 \):18/6=3 → same scale factor (3). So that's correct.

Step2: Verify Each Option

• Option a: \( g=4, h=7 \). Check ratio \( \frac{3}{4} \) vs \( \frac{6}{7} \). \( 3/4=0.75 \), \( 6/7≈0.857 \) → not equal.
• Option b: \( g=6, h=8 \). \( 3/6=0.5 \), \( 6/8=0.75 \) → not equal.
• Option c: \( g=10, h=13 \). \( 3/10=0.3 \), \( 6/13≈0.461 \) → not equal.
• Option d: \( g=9, h=18 \). \( 3/9=1/3 \), \( 6/18=1/3 \) → equal. So the ratio of corresponding sides is equal, so triangles are similar.

Answer:

d) \( g = 9, h = 18 \)