Question

what is the missing length?
14 mm
area = 112 mm²
t = millimeters
submit

Explanation:

Step1: Recall the area formula for a triangle

The area \( A \) of a triangle is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Here, the height is \( 14 \) mm, the area \( A = 112 \) \( \text{mm}^2 \), and the base is \( 2t \) (since \( t \) is half the base, as the dotted line is the altitude to the base, splitting it into two equal parts? Wait, no, actually, looking at the diagram, the dotted line is the height, and \( t \) is half the base? Wait, no, maybe the base is \( 2t \)? Wait, no, maybe the base is \( t \times 2 \)? Wait, no, let's re-examine. Wait, the triangle: the height is 14 mm, and the base is the length of the bottom side, which is split by the altitude into two parts, but maybe in this case, the base is \( 2t \)? Wait, no, maybe the base is \( t \times 2 \)? Wait, no, the formula is \( A = \frac{1}{2} \times \text{base} \times \text{height} \). Let's denote the base as \( b \), so \( 112 = \frac{1}{2} \times b \times 14 \). Wait, but in the diagram, \( t \) is half the base? Wait, no, maybe the base is \( 2t \)? Wait, no, let's solve for \( b \) first.

Step2: Solve for the base \( b \)

From \( A = \frac{1}{2} \times b \times h \), we can rearrange for \( b \): \( b = \frac{2A}{h} \). Plugging in \( A = 112 \) and \( h = 14 \): \( b = \frac{2 \times 112}{14} \). Calculate numerator: \( 2 \times 112 = 224 \). Then \( b = \frac{224}{14} = 16 \) mm. Wait, but in the diagram, \( t \) is half the base? Wait, no, maybe the base is \( 2t \), so \( t = \frac{b}{2} \)? Wait, no, looking at the diagram, the dotted line is the height, and \( t \) is the segment from the foot of the height to the vertex of the base? Wait, no, maybe the base is \( 2t \), so the base length is \( 2t \). Wait, no, let's re-express. Wait, maybe the base is \( t \times 2 \), so the formula is \( A = \frac{1}{2} \times (2t) \times 14 \). Then \( 112 = \frac{1}{2} \times 2t \times 14 \). Simplify: \( \frac{1}{2} \times 2t = t \), so \( 112 = t \times 14 \). Then \( t = \frac{112}{14} = 8 \). Ah, that makes sense. So the base is \( 2t \), so the area formula is \( A = \frac{1}{2} \times (2t) \times 14 = t \times 14 \). So solving \( 14t = 112 \), we get \( t = 8 \).

Wait, let's do it step by step. Let's use the area formula correctly. The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). Let the base be \( B \), height \( h = 14 \), area \( A = 112 \). So:

\( 112 = \frac{1}{2} \times B \times 14 \)

Multiply both sides by 2: \( 224 = B \times 14 \)

Divide both sides by 14: \( B = \frac{224}{14} = 16 \)

But in the diagram, \( t \) is half of \( B \)? Wait, no, maybe the base is \( 2t \), so \( B = 2t \), so \( 2t = 16 \), so \( t = 8 \). Yes, that's correct. So the missing length \( t \) is 8 millimeters.

Answer:

8