Question

[n the given quadrilateral \(A B C D, A B=15 \mathrm{~cm}, B C=20 \mathrm{~cm}\) and \(A D=7 \mathrm{~cm}\) \(\angle A B C=\angle A D C=90^{\circ}\) Find the length of side \(C D\) : (a) \(12 \mathrm{~cm}\) (b) \(18 \mathrm{~cm}\) (c) \(24 \mathrm{~cm}\) (d) none of the above

Short answer

Answer: (c) 24 cm

Step-by-step solution

Identify the given information in the problem

In the quadrilateral \(ABCD\), we have the following given information: - \(AB = 15 \mathrm{~cm}\) - \(BC = 20 \mathrm{~cm}\) - \(AD = 7 \mathrm{~cm}\) - \(\angle ABC = \angle ADC = 90^\circ\)

Apply the Pythagorean theorem in \(\triangle ABC\)

Since \(\triangle ABC\) is a right triangle with \(\angle ABC = 90^\circ\), we can apply the Pythagorean theorem to find the length of the side \(AC\). Let \(AC = x\). $$AB^2 + BC^2 = AC^2$$ $$15^2 + 20^2 = x^2$$ $$225 + 400 = x^2$$ $$625 = x^2$$ $$x = AC = 25 \mathrm{~cm}$$

Apply the Pythagorean theorem in \(\triangle ADC\)

Since \(\triangle ADC\) is also a right triangle with \(\angle ADC = 90^\circ\), we can apply the Pythagorean theorem to find the length of the side \(CD\). Let \(CD = y\). $$AD^2 + DC^2 = AC^2$$ $$7^2 + y^2 = 25^2$$ $$49 + y^2 = 625$$

Solve for \(y\)

We have the equation \(49 + y^2 = 625\). Now we will solve for \(y\). $$y^2 = 625 - 49$$ $$y^2 = 576$$ $$y = CD = \sqrt{576}$$ $$y = 24 \mathrm{~cm}$$

Check the answer with the given options

The length of side \(CD\) is \(24 \mathrm{~cm}\), which corresponds to option (c). Thus, the correct answer is \((c) \: 24 \mathrm{~cm}\).

Key concepts

Understanding Geometry Problems

Geometry problems often involve figures like triangles, quadrilaterals, and circles. To solve these kinds of problems, one must first thoroughly analyze the given information. In a typical geometry problem:

• Identify the type of shape you are dealing with (e.g., triangle, quadrilateral).
• Note down any given measurements or angles.
• Recognize any special properties, such as right angles or lines of symmetry.

In the problem we are discussing, we have a quadrilateral consisting of two right triangles. Key information such as side lengths and angles is provided. Utilizing this information allows us to compartmentalize the quadrilateral into more manageable sections (in this case, right triangles) to solve for any unknown sides or angles. An effective approach in geometry problems is to break down complex shapes and apply known formulas or theorems.

Insights into Quadrilateral Properties

A quadrilateral is any four-sided polygon. They come in various forms, such as rectangles, squares, parallelograms, and more.

• Angles: The total of the interior angles in a quadrilateral is always 360 degrees.
• Types: Certain quadrilaterals, like rectangles and squares, have special characteristics, including equal side lengths or angles.
• Special Triangles: Sometimes, quadrilaterals can be split into smaller components, such as triangles, which help solve complex problems more easily.

In the given problem, quadrilateral ABCD is comprised of two right triangles. This means it has two 90-degree angles. Understanding these properties allows us to apply the Pythagorean theorem effectively, making it possible to find missing side lengths.

Right Triangle Calculation using Pythagorean Theorem

The Pythagorean theorem is a fundamental tool in geometry used to calculate the lengths of sides in right triangles. The theorem states:\[ a^2 + b^2 = c^2 \]where \(c\) is the hypotenuse (the side opposite the right angle) and \(a\) and \(b\) are the other two sides.For example, in right triangle \(\triangle ABC\) given in the problem:

• Sides: \(AB = 15\, \text{cm}, BC = 20\, \text{cm}\)
• The theorem leads us to calculate the hypotenuse \(AC\).

Plugging into the theorem:\[ 15^2 + 20^2 = AC^2 \]\[ 225 + 400 = AC^2 \]\[ 625 = AC^2 \]\[ AC = 25 \, \text{cm} \]With \(AC\) known, we can similarly solve for \(CD\) in \(\triangle ADC\), knowing \(AD = 7\, \text{cm}\):\[ 7^2 + y^2 = 25^2 \]\[ y = 24 \, \text{cm} \]Thus, by breaking the problem into smaller problems, the Pythagorean theorem easily helps find unknown side lengths.