Exercises 3 & 4 – Animated Solutions ത്രികോണഭാഗം · മധ്യഭാഗം
Nine questions. The first four use "a line parallel to a side cuts the other two sides in the same ratio" (and its converse); the last five use the midpoint theorem. Read each question, try it in your notebook (use the hint if you are stuck), then press Next ▶ and watch the figure and the proof grow together.
- Key idea 5/6: a line parallel to one side of a triangle cuts the other two sides in the same ratio – into the same fractions.
- Key idea 7: a line cutting two sides in the same ratio is parallel to the third side.
- Midpoint theorem (Key ideas 8–10): the line joining the midpoints of two sides is parallel to the third side and half of it; through one midpoint, parallel to a side, you reach the other midpoint.
- Parallelogram tests (page 2): opposite sides equal and parallel; or one pair of opposite sides equal and parallel.
Exercise 3 · Question 1 – pieces of the left side
In the figure, the green line is parallel to the right side of the blue triangle. The left side is 6 cm long and the base is split into 3 cm and 5 cm. Find the lengths of the two pieces of the left side.
(ഈ വര ത്രികോണത്തിന്റെ ഇടതുവശത്തെ മുറിക്കുന്ന ഭാഗങ്ങളുടെ നീളം കണക്കാക്കുക.)
The green line is parallel to the right side, so it cuts the base and the left side into the same fractions. What fraction of the base is the 3 cm piece?
Exercise 3 · Question 2 – a parallelogram and its diagonal
In the parallelogram ABCD, the line through P on AB parallel to BC meets AC at Q, and the line through Q parallel to AB meets AD at R. Prove that (i) AP/PB = AR/RD and (ii) AP/AB = AR/AD.
The diagonal AC splits the parallelogram into two triangles, ABC and ACD. PQ is parallel to a side of the first; QR is parallel to a side of the second (why?). Apply Key idea 5 in each and compare.
Exercise 3 · Question 3 – trisecting a diagonal
In the figure, two opposite vertices of a parallelogram are joined to the midpoints of two opposite sides. Prove that these two lines divide the diagonal into three equal parts.
First show that the two lines are parallel (DMBN is itself a parallelogram). Then look at two triangles, each having one of the lines through the midpoint of a side – a 1 : 1 ratio.
Exercise 3 · Question 4 – two parallels in one triangle
In triangle ABC, the line through P on BC parallel to AC meets AB at Q, and the line through Q parallel to AP meets BC at R. Prove that BP/PC = BR/RP.
Both ratios can be written as BQ/QA – one from triangle ABC, the other from triangle ABP.
Exercise 4 · Question 1 – the midpoint triangle
In the figure, the midpoints of the sides of a triangle are joined to make another triangle. How many times the perimeter of the small triangle is the perimeter of the big one? And the area?
(വലിയ ത്രികോണത്തിന്റെ ചുറ്റളവ്, ചെറിയ ത്രികോണത്തിന്റെ ചുറ്റളവിന്റെ എത്ര മടങ്ങാണ്? പരപ്പളവോ?)
Each side of the small triangle is half a side of the big one. And the three midlines cut the big triangle into four congruent pieces.
Exercise 4 · Question 2 – cutting out the middle
A paper triangle (first picture). The middle triangle formed by the midpoints is cut out (second picture). From each of the three remaining triangles the middle triangle is again cut out (third picture). (i) What fraction of the first picture's area is the second picture? (ii) And the third?
Cutting out the middle triangle removes ¼ and leaves ¾. Doing it again to every remaining piece leaves ¾ of ¾.
Red is paper, white is cut away. Each cut keeps ¾ of what was there.
Exercise 4 · Question 3 – a rectangle inside a right triangle
A quadrilateral is drawn with the right-angle vertex of a right triangle and the midpoints of its sides as vertices. (i) Prove that it is a rectangle. (ii) What fraction of the triangle's area is the rectangle's area?
Two of its sides are midlines of the triangle – so they are parallel to the legs and half as long. A parallelogram with a right angle is a rectangle.
Exercise 4 · Question 4 – two triangles on one line
Two points above a line are joined to its ends, making two triangles. The midpoints of the left and right sides of the two triangles are joined to make a quadrilateral. (i) Prove that it is a parallelogram. (ii) Where should the two points be for it to be (a) a rhombus (b) a rectangle (c) a square? (iii) Is everything still true if one point is above the line and the other below?
Two of the four sides are midlines of triangles standing on AB – both parallel to AB and half of it. The other two are midlines of triangles with PQ as a side.
Try to make PQ equal to AB (rhombus), PQ perpendicular to AB (rectangle), or both (square). Drag Q below the line – the quadrilateral is still a parallelogram.
Exercise 4 · Question 5 – midpoints of any quadrilateral
(i) Prove that the quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram. (ii) What must the original quadrilateral be like for the inner one to be (a) a rhombus (b) a rectangle (c) a square?
Draw one diagonal: it splits the quadrilateral into two triangles, and two of the midlines are parallel to that diagonal and half of it. The other diagonal does the same for the other two sides.
Make the diagonals equal (rhombus), perpendicular (rectangle) or both (square). Even a "dented" quadrilateral gives a parallelogram.
Check yourself
- To use the ratio theorem, first name the triangle and the side the line is parallel to; then write the two equal ratios.
- A ratio that appears in two different triangles (like BQ/QA) is a bridge between two other ratios.
- "Midpoint + parallel" always means: the midline is parallel to the third side and half of it – the key to every parallelogram question here.
- Repeatedly cutting out the middle triangle keeps ¾ each time: ¾, 9/16, 27/64, …
Next: Section 5 – the altitudes and medians of a triangle meet at one point ▶