Question

1. a triangle has a height of 14 cm with an area of 105 cm². if the area remains the same, what is the height if the base is doubled?

options: 28 cm, 14 cm, 7 cm, 7.5 cm

Explanation:

Step1: Recall triangle area formula

The area of a triangle is given by \( A = \frac{1}{2} \times b \times h \), where \( b \) is the base and \( h \) is the height. We know \( A = 105 \, \text{cm}^2 \) and \( h = 14 \, \text{cm} \) (assuming a typo, since 15 might be 14? Wait, original height is 14? Wait, the problem says height of 14 cm? Wait, the user's problem: "A triangle has a height of 14 cm with an area of 105 cm². If the area remains the same, what is the height if the base is doubled?" Wait, maybe the original height is 14. Let's proceed. First, find the original base.

From \( A = \frac{1}{2}bh \), solve for \( b \): \( b = \frac{2A}{h} \). Substitute \( A = 105 \), \( h = 14 \): \( b = \frac{2 \times 105}{14} = \frac{210}{14} = 15 \, \text{cm} \).

Step2: Double the base and find new height

New base \( b' = 2 \times 15 = 30 \, \text{cm} \). Area remains \( A = 105 \, \text{cm}^2 \). Use the area formula again: \( A = \frac{1}{2}b'h' \), solve for \( h' \): \( h' = \frac{2A}{b'} \). Substitute \( A = 105 \), \( b' = 30 \): \( h' = \frac{2 \times 105}{30} = \frac{210}{30} = 7 \, \text{cm} \)? Wait, no, wait, maybe the original height is 14, but let's check again. Wait, maybe the original height is 14, area 105. Wait, maybe I made a mistake. Wait, let's re-express.

Wait, the problem: "A triangle has a height of 14 cm with an area of 105 cm². If the area remains the same, what is the height if the base is doubled?"

Original: \( A = \frac{1}{2}bh \implies 105 = \frac{1}{2} \times b \times 14 \implies 105 = 7b \implies b = 15 \, \text{cm} \).

New base: \( 2b = 30 \, \text{cm} \). Area still 105. So \( 105 = \frac{1}{2} \times 30 \times h' \implies 105 = 15h' \implies h' = 7 \, \text{cm} \)? But the options include 7 cm. Wait, the options are 28, 14, 7, 7.5? Wait, maybe the original height is 14, but let's check again. Wait, maybe the original height is 14, area 105. Then base is 15. Double base to 30, area 105, so height is 7. So the correct option is 7 cm.

Wait, maybe the original problem had height 14, not 15. Let's confirm. So the steps:

1. Find original base: \( b = \frac{2A}{h} = \frac{2*105}{14} = 15 \).

1. New base: 30.

1. New height: \( h' = \frac{2A}{b'} = \frac{2*105}{30} = 7 \).

Answer:

The correct option is the one with 7 cm (the third option, e.g., C. 7 cm if options are labeled, but based on the options given, the one with 7 cm).