Math Problem Statement
Solution
The uploaded image describes a geometry problem involving a trapezoid ABCD, with the following information:
- ∠A = ∠D = 90°,
- AB = 10 cm, CD = 30 cm, and AD = 35 cm,
- Point M is on side AD, and AM = 15 cm.
The tasks are:
- Prove that triangles ABM and DMC are similar.
- Prove that ∠BMC = 90°.
I will now solve the problem step by step.
Part (a): Prove that △ABM ∼ △DMC
To show that two triangles are similar, we need to prove that they satisfy one of the criteria for similarity (such as AA - Angle-Angle, SAS - Side-Angle-Side, or SSS - Side-Side-Side).
- First, we notice that both triangles share the angle ∠BAM, as both triangles are formed by the same line segments AM.
- Next, since ∠A = ∠D = 90° by given information, we can conclude that ∠ABM = ∠DMC = 90°.
Since we now have two corresponding angles that are equal (∠BAM = ∠DMC and ∠ABM = ∠DMC), by AA similarity criterion, we can conclude that △ABM ∼ △DMC.
Part (b): Prove that ∠BMC = 90°
To prove that ∠BMC = 90°, consider the following:
- Since △ABM ∼ △DMC, the corresponding sides are proportional.
- Now, notice that in both triangles, the angle ∠BAM is common, and both triangles share a right angle at ∠ABM and ∠DMC.
- The sum of angles in any triangle is 180°, so considering the angle sum property and the geometry of the figure, we can deduce that ∠BMC must be a right angle, as it is the complement of ∠BAM and ∠DMC.
Thus, ∠BMC = 90°.
Would you like further details or clarification on any part of this solution?
Related questions:
- How can you prove similarity in triangles using the SAS criterion?
- What is the geometric significance of the perpendicular lines in this trapezoid?
- How does the angle sum property apply to similar triangles?
- What are other ways to prove two triangles are similar apart from AA?
- How does proportionality of sides relate to similarity in triangles?
Tip:
In geometry, when working with similar triangles, always check for common angles and proportional sides, as they are often the key to proving similarity.
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Math Problem Analysis
Mathematical Concepts
Geometry
Triangle Similarity
Right Angles
Formulas
AA Criterion for Similarity
Angle Sum in Triangle
Theorems
AA Similarity Criterion
Angle Sum Property
Suitable Grade Level
Grades 9-11
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