Math Problem Statement

Calculate the area of triangle ABC, work out the length of BX, and use the given area of triangle DCX to find the area of triangle ABX.

Solution

Let's go through the mathematical questions visible in the image and break them down:

Part (a)

"Complete the statement."

The problem shows two triangles: triangle ABXABX and triangle DCXDCX. Since ABCDAB \parallel CD, and the triangles share angle XX, by AA (Angle-Angle) similarity criterion, we can say:

Answer:

  • Triangle ABXABX is similar to triangle DCXDCX.

Part (b)

"Work out the length of BXBX."

We are given that ABCDAB \parallel CD and that the lengths are:

  • AB=6cmAB = 6 \, \text{cm}
  • CD=12cmCD = 12 \, \text{cm}
  • CX=8cmCX = 8 \, \text{cm}
  • DX=7cmDX = 7 \, \text{cm}

Using the similarity of triangles ABXABX and DCXDCX, we can set up the proportion:

ABCD=BXCX\frac{AB}{CD} = \frac{BX}{CX}

Substitute the known values:

612=BX8\frac{6}{12} = \frac{BX}{8}

Simplifying the equation:

12=BX8\frac{1}{2} = \frac{BX}{8}

Multiply both sides by 8 to find BXBX:

BX=4cmBX = 4 \, \text{cm}

Answer:

  • BX=4cmBX = 4 \, \text{cm}

Part (c)

"The area of triangle DCXDCX is 26.906 cm². Use this value to find the area of triangle ABXABX."

Since the triangles ABXABX and DCXDCX are similar, the ratio of their areas is proportional to the square of the ratio of their corresponding sides.

Area of ABXArea of DCX=(ABCD)2\frac{\text{Area of } ABX}{\text{Area of } DCX} = \left(\frac{AB}{CD}\right)^2

Substitute the known values:

Area of ABX26.906=(612)2\frac{\text{Area of } ABX}{26.906} = \left(\frac{6}{12}\right)^2

Simplify:

Area of ABX26.906=14\frac{\text{Area of } ABX}{26.906} = \frac{1}{4}

Multiply both sides by 26.906 to find the area of ABXABX:

Area of ABX=26.9064=6.7265cm2\text{Area of } ABX = \frac{26.906}{4} = 6.7265 \, \text{cm}^2

Answer:

  • Area of triangle ABXABX = 6.73 cm² (rounded to 2 decimal places).

Do you want further details on any of these steps?
Here are some questions you might find useful:

  1. How do you apply the AA similarity criterion in triangles?
  2. Can you explain the proportionality rule for corresponding sides in similar triangles?
  3. How do you compute the area ratio for similar triangles?
  4. What other methods could be used to calculate the length of BXBX?
  5. What is the geometric meaning behind the similarity of these triangles?

Tip: When working with similar triangles, always check if you can set up proportional relationships between corresponding sides for easy calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Similarity
Proportions
Area of Triangles

Formulas

Proportionality: (AB/CD) = (BX/CX)
Area ratio of similar triangles: (Area of ABX / Area of DCX) = (AB/CD)^2

Theorems

AA Similarity Criterion
Proportionality in Similar Triangles

Suitable Grade Level

Grades 9-11