Calculate Triangle Fit in Deltoid: 8cm and 14cm Dimensions Problem

The deltoid ABCD is shown below.

Given in cm:

AC = 8

DB = 14

Calculate how many times triangle ABC fits into the deltoid ABCD.

Video Solution

Solution Steps

00:00 Calculate how many times triangle ABC fits in quadrilateral ABCD

00:03 The area of the quadrilateral equals the sum of the areas of triangles ABC and ACD

00:24 Let's prove triangle congruence

00:28 They share the same side (diagonal AC) (side)

00:33 The main diagonal in the quadrilateral bisects the vertex angle (angle)

00:48 Pair of equal sides in the quadrilateral (side)

01:12 Therefore triangles are congruent according to SAS

01:25 Now let's find the area ratio between the quadrilateral and triangle

01:29 Due to the congruence it fits twice, and that's the answer to the question

Step-by-Step Solution

To solve this problem, we'll calculate the areas to determine how many times $\triangle ABC$ fits into $ABCD$ :

Step 1: Calculate the area of the deltoid $ABCD$ using the formula:
$\text{Area of deltoid} = \frac{1}{2} \times AC \times DB = \frac{1}{2} \times 8 \times 14 = 56 \, \text{square cm}$
Step 2: Calculate the area of $\triangle ABC$ :
$\triangle ABC$ has diagonal $AC = 8 \, \text{cm}$ , so $\text{Area of} \, \triangle ABC = \frac{1}{2} \times AC \times DB = \frac{1}{2} \times 8 \times \frac{14}{2} = 28 \, \text{square cm}$
Step 3: Determine how many times $\triangle ABC$ fits into deltoid $ABCD$ :
Since the area of the deltoid is $56 \, \text{square cm}$ and the area of $\triangle ABC$ is $28 \, \text{square cm}$ , then:
$\frac{\text{Area of deltoid}}{\text{Area of} \, \triangle ABC} = \frac{56}{28} = 2$

Therefore, triangle $\triangle ABC$ fits $\textbf{2}$ times into the deltoid $ABCD$ .