Triangle Relationships review games for Geometry.
This unit covers midsegments, triangle inequality and angle bisectors and medians — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers midsegments, triangle inequality and angle bisectors and medians — essential concepts for Geometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Midsegments
A midsegment connects the midpoints of two sides of a triangle. It is always parallel to the third side and exactly half its length. Exams test both the length relationship and the parallel relationship.
Key Points
- Midsegment length = ½ × the length of the parallel side
- The midsegment is parallel to the third side (not connected to it)
- A triangle has exactly 3 midsegments
- Setting up and solving equations using the ½ relationship is the most common exam task
In triangle ABC, D is the midpoint of AB and E is the midpoint of AC. If DE = 3x + 1 and BC = 8x − 6, find DE.
By the Midsegment Theorem, DE = ½ · BC, so 3x + 1 = ½(8x − 6). Multiply both sides by 2: 6x + 2 = 8x − 6, giving x = 4. Substitute back: DE = 3(4) + 1 = 13.
2 Triangle Inequality
The Triangle Inequality Theorem states that the sum of any two side lengths of a triangle must be greater than the third side. Exams ask you to determine whether three lengths can form a triangle or to find the range of possible values for a missing side.
Key Points
- For sides a, b, c: a + b > c, a + c > b, and b + c > a must ALL be true
- The most efficient check: if the sum of the two smaller sides is greater than the largest, all three conditions are met
- For a missing side x: |a − b| < x < a + b
- The longest side is always opposite the largest angle
Two sides of a triangle measure 7 and 11. Find all possible integer values for the third side x.
Apply the range formula: |11 − 7| < x < 11 + 7, which simplifies to 4 < x < 18. The third side must be strictly between 4 and 18, so the possible integer values are 5, 6, 7, …, 17.
3 Angle Bisectors
An angle bisector divides an angle into two equal halves and intersects the opposite side. The three angle bisectors of a triangle meet at the incenter, which is equidistant from all three sides. The Angle Bisector Theorem states that the bisector divides the opposite side proportionally to the two adjacent sides.
Key Points
- Angle Bisector Theorem: if BD bisects angle B in triangle ABC, then AD/DC = AB/BC
- The incenter is the point equidistant from all three sides (center of the inscribed circle)
- The incenter is always inside the triangle
- Exams most commonly test setting up and solving the proportional segments equation
In triangle ABC, BD bisects angle B. If AB = 10, BC = 6, and AC = 8, find AD and DC.
By the Angle Bisector Theorem, AD/DC = AB/BC = 10/6 = 5/3. Since AD + DC = AC = 8, set AD = 5k and DC = 3k, so 5k + 3k = 8, giving k = 1. Therefore AD = 5 and DC = 3.
4 Medians
A median connects a vertex to the midpoint of the opposite side. The three medians meet at the centroid, which divides each median in a 2:1 ratio from vertex to midpoint. Exams focus almost entirely on applying this 2:1 ratio to find segment lengths.
Key Points
- The centroid divides each median so that the vertex-to-centroid segment is twice the centroid-to-midpoint segment
- If G is the centroid and M is the midpoint, then vertex-to-G = (2/3) of the full median, G-to-M = (1/3) of the full median
- The centroid is always inside the triangle
- A triangle has exactly 3 medians, each going from a vertex to the opposite side's midpoint
Median AM has a total length of 18. G is the centroid. Find AG and GM.
The centroid divides the median in a 2:1 ratio from the vertex. So AG = (2/3)(18) = 12 and GM = (1/3)(18) = 6. Always assign the larger piece to the vertex side.
Questions, answered.
What is Triangle Relationships?
Triangle Relationships is Unit 5 of Geometry, covering midsegments, triangle inequality and angle bisectors and medians.
How to study for Geometry Unit 5?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.