Segments, perpendicular height and symmetry
Use a stated scale and segment order, choose the perpendicular height of a triangle, and justify symmetry claims.
What you will tackle
- Estimate and combine collinear segment lengths
- Use triangle area with perpendicular height
- Use isosceles base angles
- Count reflection axes without assuming every quadrilateral is regular
Read what a diagram guarantees
For collinear A,B,C in that order, AC = AB + BC. Shared or overlapping segments can be found by subtracting known totals. Estimate from a picture only when its scale is stated; drawn proportions are otherwise insufficient. Triangle area is bh/2, where the height is perpendicular to the chosen base, sometimes outside the triangle.
Equal sides and reflection
An isosceles triangle's equal sides face equal base angles. Its line from apex to base midpoint is a reflection axis. An equilateral triangle has three reflection axes; a nonsquare rectangle has two; a square has four; a circle has infinitely many. A generic parallelogram has no reflection axis even though it has half-turn symmetry.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
On the displayed scale segment A–B–C–D, AD = 10 and the marks show AB = 3 and AC = 7. Find BC and BD. A triangle has base 8 cm and perpendicular height 5 cm. Find its area.
Show a starting hint
Identify which whole segment contains each smaller one.
Compare with the explained answer
- BC = AC − AB = 7 − 3 = 4.
- BD = AD − AB = 10 − 3 = 7.
- Triangle area is 8·5/2 = 20 cm²; a sloping side cannot replace the perpendicular height.
Answer / response criteria: BC 4; BD 7; area 20 cm².
Independent practice
Independent 1
In a non-equilateral isosceles triangle, the apex angle is 40°. Find its base angles and number of reflection axes. Give the reflection-axis counts of a nonsquare rectangle, a square and a generic parallelogram.
Compare with the explained answer
- The equal base angles sum to 140°, so each is 70°.
- The isosceles triangle has one reflection axis through apex and base midpoint.
- The axis counts are 2, 4 and 0 respectively; a half-turn is a different symmetry.
Answer / response criteria: 70°,70°; one axis; 2,4,0.
Application practice
Application 1
A triangular sign has area 30 m² and base 12 m. Find its perpendicular height. A to-scale diagram shows a reference segment of 10 units and another segment 60% as long; estimate the second length. Explain why the estimate would fail without a scale guarantee.
Compare with the explained answer
- From 30 = 12h/2, h = 5 m.
- The stated ratio gives 0.6·10 = 6 units.
- Without a scale guarantee, a drawn length ratio need not equal the mathematical length ratio, so only labelled facts could justify an exact result.
Answer / response criteria: 5 m; about 6 units; an unstated drawing scale is insufficient.
What can you explain now?
A triangular sign has area 30 m² and base 12 m. Find its perpendicular height. A to-scale diagram shows a reference segment of 10 units and another segment 60% as long; estimate the second length. Explain why the estimate would fail without a scale guarantee.
Compare with the explanation
5 m; about 6 units; an unstated drawing scale is insufficient.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide