Rectangle Area Problem: Using 25-Unit Diagonal and 7-Unit Width

Pythagorean Theorem with Diagonal Applications

The rectangle ABCD is shown below.

BD=25,BC=7 BD=25,BC=7

Calculate the area of the rectangle.

AAABBBCCCDDD725

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Calculate the area of rectangle ABCD
00:04 We'll use the Pythagorean theorem in triangle BCD
00:10 We'll substitute appropriate values and solve for DC
00:18 We'll isolate DC
00:30 This is the length of DC, now that we have the sides we can calculate the area
00:37 Multiply length(7) by width(24)
00:44 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

The rectangle ABCD is shown below.

BD=25,BC=7 BD=25,BC=7

Calculate the area of the rectangle.

AAABBBCCCDDD725

2

Step-by-step solution

We will use the Pythagorean theorem in order to find the side DC:

(BC)2+(DC)2=(DB)2 (BC)^2+(DC)^2=(DB)^2

We begin by inserting the existing data into the theorem:

72+(DC)2=252 7^2+(DC)^2=25^2

49+DC2=625 49+DC^2=625

DC2=62549=576 DC^2=625-49=576

Finally we extract the root:

DC=576=24 DC=\sqrt{576}=24

3

Final Answer

168

Key Points to Remember

Essential concepts to master this topic
  • Rectangle Property: Diagonal creates right triangle with opposite sides
  • Technique: Use a2+b2=c2 a^2 + b^2 = c^2 where c is diagonal
  • Check: Verify 72+242=49+576=625=252 7^2 + 24^2 = 49 + 576 = 625 = 25^2

Common Mistakes

Avoid these frequent errors
  • Using diagonal as a side length in area calculation
    Don't calculate area as 25 × 7 = 175! The diagonal is NOT a side of the rectangle. This gives the wrong area because diagonals don't form the rectangle's perimeter. Always find both perpendicular sides first, then multiply length × width.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

1.51.51.5AAABBBCCCDDD9.5

FAQ

Everything you need to know about this question

Why can't I just multiply the diagonal by the given side?

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The diagonal is the hypotenuse of a right triangle, not a side of the rectangle! Rectangle area requires length × width, which are the two perpendicular sides.

How do I know which sides to use in the Pythagorean theorem?

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In a rectangle, the diagonal is always the longest side (hypotenuse). The two given measurements are the legs of the right triangle formed by the diagonal.

What if I get a decimal when finding the missing side?

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That's normal! Use a calculator to find 576=24 \sqrt{576} = 24 . Some problems have perfect square roots, others don't. Keep all decimal places for accuracy.

Can I use this method for any quadrilateral?

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No! This only works for rectangles because they have right angles. Other quadrilaterals don't form right triangles with their diagonals.

How do I check if my area calculation is reasonable?

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Compare your answer to simple estimates: if sides are about 7 and 24, the area should be around 7 × 24 = 168. This matches our exact answer!

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