[cite: 3, 6]
The Synthesis of Rules
In Mastery Assessment II, we are breaking down the walls between geometry and algebra. [cite: 6] We saw earlier that triangles with the same area are not necessarily congruent because you can stretch the base and shrink the height. [cite: 3]
But what if we combine that geometric rule with an algebraic constraint? What if we force the triangle to be equilateral? Suddenly, the flexible base and height are locked together by an algebraic equation. Will the false converse suddenly become true? It is time to chain our logical rules together!
In Mastery Assessment II, we are breaking down the walls between geometry and algebra. [cite: 6] We saw earlier that triangles with the same area are not necessarily congruent because you can stretch the base and shrink the height. [cite: 3]
But what if we combine that geometric rule with an algebraic constraint? What if we force the triangle to be equilateral? Suddenly, the flexible base and height are locked together by an algebraic equation. Will the false converse suddenly become true? It is time to chain our logical rules together!
1 Equilateral Area Logic
2 Chained Algebraic Roots
Objective 1: Re-evaluating the Square Trap
Objective 2: The Altitude Center
In Question 3, we defined the incentre as the intersection of angle bisectors. What geometric center is formed if we instead drop altitudes (perpendicular heights) from all three vertices to the opposite sides? [cite: 6]
Concept:
1
A triangle has four primary "centers" based on different sets of intersecting lines: Medians (Centroid), Angle Bisectors (Incentre), Perpendicular Bisectors (Circumcentre), and Altitudes (Orthocentre). [cite: 6]
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Hint:
1
"Ortho" comes from the Greek word for "straight" or "right angle". Which set of lines explicitly drops down at a right angle (perpendicular) from the vertex? The altitudes!
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Mistakes to Avoid:
!
Confusing altitudes with perpendicular bisectors: An altitude drops from a vertex and hits the opposite side at 90 degrees. A perpendicular bisector hits the side at 90 degrees, but it must hit exactly in the middle, and it doesn't necessarily pass through the opposite vertex!
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Solution:
The textbook defines the incentre as the intersection of the angle bisectors. [cite: 6]
By definition in geometry, the intersection of the three altitudes (the lines drawn from a vertex perpendicular to the opposite side) is called the Orthocentre.
Correct Option: (D)
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Extra Practice 1
