Answer:
4.8 cm
Step-by-step explanation:
Quadrilateral ABCD is made up of two right triangles, ΔABC and ΔDAC.
Triangle ABC has BC as its base and AB as its height.
Triangle DAC has AC as its base and AD as its height.
The area of any triangle is half the product of its base and height.
Therefore, the area of quadrilateral ABCD can be expressed as:
[tex]\textsf{Area $ABCD$}=\dfrac{1}{2}\cdot BC \cdot AB + \dfrac{1}{2}\cdot AC \cdot AD[/tex]
We have been given the area of the quadrilateral and the lengths of BC and AC. Therefore, to find AD, we must first find the length of AB. To do this, use the Pythagorean Theorem:
[tex]AB^2+BC^2=AC^2\\\\AB^2+6^2=7.5^2\\\\AB^2 + 36 = 56.25\\\\AB^2=20.25\\\\AB=\sqrt{20.25}\\\\AB=4.5\; \rm cm[/tex]
Therefore:
- AB = 4.5 cm
- BC = 6 cm
- AC = 7.5 cm
- Area ABCD = 31.5 cm²
Substitute these values into the area equation and solve for AD:
[tex]\dfrac{1}{2}\cdot 6 \cdot 4.5 + \dfrac{1}{2}\cdot 7.5 \cdot AD=31.5\\\\\\3 \cdot 4.5 + 3.75 \cdot AD=31.5\\\\\\13.5 + 3.75 \cdot AD=31.5\\\\\\3.75 \cdot AD=18\\\\\\AD=\dfrac{18}{3.75}\\\\\\AD=4.8\; \rm cm[/tex]
Therefore, the length of AD is:
[tex]\Large\boxed{\boxed{AD=4.8\; \rm cm}}[/tex]
