Math exercises on the Thales theorem in the third grade are to be treated seriously. Thus, these exercises for third graders in middle school focus on the direct and reciprocal part of Thales’ theorem.
Questions 2, 3 and 4 are independent. The unit is the centimeter.
1) Construct a triangle MAI rectangular in A such that AM = 8 and IM = 12. Briefly outline the stages of construction.
2) Calculate the exact value of AI.
3) R is the point on the segment [MI] such that MR = 9.
The parallel to (AI) passing through R intersects [AM] at E.
4) Calculate .
Deduct the value, rounded to the nearest degree, of .
Let be a triangle ADE rectangle in A such that :
AD = 5 cm and AE = 3 cm.
B is the point on the half-line [AD) such that BA = 8 cm.
The parallel to the line (DE) through B intersects (AE) at C.
1) Make the figure.
2) Calculate DE. Give a value rounded to the nearest mm.
3) Calculate AC.
4) Calculate BC. Give a value rounded to the nearest mm.
5) Calculate .
6) Deduce the measure of the angle rounded to the degree.
Let IJK be a right triangle in I such that IJ = 3.6 cm and IK = 4.8 cm.
We place the point L of the half-line [KI) such that KL = 8 cm.
The parallel to the line (IJ) passing through L cuts (KJ) at M.
The figure on the right is not full size and should not be reproduced.
1) Show that KJ = 6 cm.
2) Calculate the value of KM, justifying the answer.
3) Determine a measure of the angle to the nearest 1 degree.
Consider the triangle ABC rectangle in A such that AB = 5, BC = 9, the unit being the cm.
1) Construct the triangle ABC in full size.
2) Calculate the exact value of AC.
3) Calculate the measure of the angle to the nearest degree by default.
4) The circle with center B and radius AB intersects the segment [BC] at M. The parallel to the line (AC) that passes through M intersects the segment [AB] at N.
- Complete the figure.
- Calculate the exact value of BN.
Exercise 5: Thales’ theorem
A neon sign manufacturer needs to make the letter z (out of welded glass tubes) to mount on the top of a storefront. Here is the diagram showing the shape and some dimensions of the sign:
The lines (AD) and (BC) intersect at O.
- Knowing that the lines (AB) and (CD) are parallel, calculate the lengths AB and OB (give the results in fractional form).
- Show that the tube [BC] is perpendicular to the line (AD).
- Calculate .
Deduce the value of the angle to the nearest degree.
Exercise 6: Thales’ theorem
Let ABC be a triangle such that: AB = 4.5 cm BC = 7.5 cm AC = 6 cm
- Construct such a triangle.
- Show that triangle ABC is right-angled.
- Calculate to the nearest degree the angle .
- M is the point on the segment [AB] such that AM = 1.5 cm, and N is the point on the segment [AC] such that NC = 4 cm.
Are the lines (MN) and (BC) parallel? Justify.
Exercise 7: Thales’ theorem
The unit is the centimeter.
- Construct a triangle RST such that: RS = 4.5 ST = 6 RT = 7.5
We will leave the construction lines.
- Show that the triangle RST is rectangular.
- a) Draw the circle (C) with center R and radius 4.5. The circle (C) intersects the segment [RT] at K.
- b) Draw the line d passing through the point K and parallel to the line (RS).
this line d intersects the segment [TS] at a point L.
Place this point on the figure.
- c) Calculate KL.
- Calculate the angle (rounding to the nearest degree will be given).
Construct the circle (C) with center O and radius 4 cm. Draw a diameter [AB] of this circle.
Construct the symmetrical point S of point O with respect to point A, then the circle (C’) of diameter [OS]. The circle (C’) intersects the circle (C) at two points T and T’.
- a) Prove that the triangle SOT is right-angled at T
- b) What does the line (ST) represent for the circle (C) ? Justify.
- Determine the measure of the angle .
- The line passing through B and parallel to the line (OT) intersects the line (ST) at P
- a) Construct the line (BP).
- b) Calculate BP
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