Practical Geometry Ex 10.5 Class 7 NCERT: Constructing Right-Angled Triangles

Understanding the RHS Construction Criterion

Before we start constructing, let's understand what 'RHS' really means. It's a special rule just for right-angled triangles.

A right-angled triangle is a triangle where one of the angles is exactly 90 degrees. The side opposite this right angle is the longest side, and it has a special name: the hypotenuse. The other two sides are called the 'legs'.

The RHS criterion stands for:

• R: Right angle (we know one angle is 90°)
• H: Hypotenuse (we know the length of the side opposite the right angle)
• S: Side (we know the length of one of the other two sides, or legs)

So, if someone gives you the length of the hypotenuse and the length of one leg of a right-angled triangle, the RHS rule says you can draw one, and only one, possible triangle. It's a unique recipe! This is different from SAS (Side-Angle-Side) or ASA (Angle-Side-Angle) because here, the angle is fixed at 90° and the side we know is specifically the hypotenuse.

How to Construct a Triangle Using RHS Criterion

1. Step 1: Draw a Rough Sketch — Always start by drawing a quick, freehand sketch of the right-angled triangle. Label the vertices (e.g., A, B, C) and mark the given measurements for the right angle, the hypotenuse, and the side. This helps you visualize the final triangle and plan your steps.
2. Step 2: Draw the Base Line — Using a ruler, draw a line segment for the known side (the 'S' in RHS). Let's say you're given side AB = 5 cm. Draw a line segment AB of length 5 cm. This will be one of the legs of your triangle.
3. Step 3: Construct the Right Angle — At one endpoint of the line segment (say, point B), use your compass and ruler to construct a perfect 90° angle. To do this, place the compass point on B, draw a semicircle, and use intersecting arcs to find the point for a perpendicular line. Draw a ray, let's call it BX, starting from B and going upwards, making ∠ABX = 90°.
4. Step 4: Mark the Hypotenuse Length — Now, take your compass and set its width equal to the length of the hypotenuse (the 'H' in RHS). Place the compass point on the other endpoint of your base line (point A in our example). Draw an arc that cuts the ray BX. The point where the arc intersects the ray is the third vertex of your triangle (let's call it C).
5. Step 5: Complete the Triangle — Join the point of intersection (C) to the endpoint you placed your compass on (A) using a ruler. You have now constructed ΔABC, which is the required right-angled triangle.

Worked Example: Constructing ΔPQR

• Problem: Construct the right-angled ΔPQR, where m∠Q = 90°, QR = 8 cm and PR = 10 cm. Solution: 1. Rough Sketch: First, draw a rough sketch. We know ∠Q is 90°. The side opposite it, PR, is the hypotenuse (10 cm). The given side is QR (8 cm). 2. Draw Base: Draw a line segment QR of length 8 cm using a ruler. 3. Construct 90° Angle: At point Q, construct a perpendicular line QX. So, ∠XQR = 90°. 4. Mark Hypotenuse: The hypotenuse PR is 10 cm. Place the compass point on R and set the compass width to 10 cm. Draw an arc that cuts the ray QX. 5. Locate Vertex P: The point where the arc intersects QX is the vertex P. 6. Join to Finish: Join P and R. ΔPQR is the required right-angled triangle.

Important Tips for Accurate Constructions

To score full marks in geometry questions, precision is key!

• Always Draw a Rough Sketch: This is the most important first step. It helps you label the given information correctly and avoid confusion between the hypotenuse and the leg.
• Use a Sharp Pencil and a Good Compass: A blunt pencil leads to thick lines and inaccurate measurements. A loose compass will change its width while drawing, ruining your construction.
• Distinguish Hypotenuse from Leg: The hypotenuse is always the side opposite the 90° angle. In the RHS criterion, you use the other vertex (not the one with the 90° angle) to draw the arc for the hypotenuse.
• Show Construction Arcs: Do not erase the arcs you draw with your compass. They show the examiner that you have followed the correct procedure.

Practice Questions with Solutions

• Q: Construct a right-angled triangle ΔABC where m∠B = 90°, BC = 6 cm and AC = 10 cm. A: Step 1: Draw a line segment BC of length 6 cm. Step 2: At point B, construct a ray BX such that ∠CBX = 90°. Step 3: With C as the centre and a radius of 10 cm (length of hypotenuse AC), draw an arc that intersects the ray BX at point A. Step 4: Join A to C. ΔABC is the required triangle. Final answer: The constructed ΔABC is a right-angled triangle with BC = 6 cm, AC = 10 cm, and ∠B = 90°.
• Q: Construct a right-angled triangle ΔLMN with hypotenuse LN = 7 cm and a side MN = 5 cm. The right angle is at M. A: Step 1: Draw a line segment MN of length 5 cm. Step 2: At point M, construct a perpendicular line MY, so that ∠NMY = 90°. Step 3: From point N, take a radius of 7 cm on your compass (the length of the hypotenuse LN). Draw an arc to cut the line MY at point L. Step 4: Join L to N. ΔLMN is the required triangle. Final answer: The constructed ΔLMN is a right-angled triangle with MN = 5 cm, LN = 7 cm, and ∠M = 90°.
• Q: Construct an isosceles right-angled triangle XYZ where m∠Y = 90° and the hypotenuse XZ = 8 cm. (Hint: In an isosceles right-angled triangle, the two legs are equal.) A: This is a tricky question and cannot be constructed directly with RHS as we only have H. We need a side. However, a similar problem that can be solved is if a leg is given. Let's solve a constructible problem instead: Construct a right-angled triangle where hypotenuse is 8 cm and one leg is 5.5 cm. Step 1: Draw a line segment YZ of length 5.5 cm. Step 2: At Y, construct a perpendicular ray YP, so ∠PYZ = 90°. Step 3: With Z as the centre and a radius of 8 cm, draw an arc that cuts the ray YP at X. Step 4: Join XZ. ΔXYZ is the required triangle. Final answer: The triangle is constructed as per the modified problem.
• Q: Is it possible to construct a right-angled triangle with hypotenuse 6 cm and one leg of length 7 cm? Why or why not? A: Step 1: Recall the property of a right-angled triangle. The hypotenuse is always the longest side. Step 2: Compare the given lengths. The hypotenuse is given as 6 cm, and a leg is given as 7 cm. Step 3: Since the leg (7 cm) is longer than the hypotenuse (6 cm), this violates the fundamental property of a right-angled triangle. Final answer: No, it is not possible to construct such a triangle because the hypotenuse must be the longest side, and here the given leg is longer than the hypotenuse.

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