grade 10 mathematics lesson ppt

1. (M10GE-IIg-2)
Session 5
MARINO R. CUDIA
Master Teacher I, SMVIHS
Applies the distance
formula to prove some
geometric properties.
DIVISION WEBINAR
ON
Enhanced Mathematics
Teaching And Learning
Program (EMTAP)

2. OBJECTIVES:
At the end of the session, the participants should
be able to:
a. recall the distance and midpoint formula,
b. applies the distance formula to prove some
geometric properties, and
c. solve problems involving geometric figures on the
rectangular coordinate plane with accuracy.

3. 3-Pics 1-Word
DISTANCE

4. 3-Pics 1-Word
PLOT

5. 3-Pics 1-Word
COORDINATES

6. Coordinate Plane: Plotting of points/Identifying the coordinates of a point
Try This!
Use the figure at the right then perform the
following:
1. Plot the following points on a coordinate
plane:
A(–4, 2), B(2, 2), C (–4, –2), and D(2, –2)
2. What is the distance of A from B?
D from C?
3. What is the distance of A from C?
B from D?
4. Compare the distances.
5. What is the figure formed?
A (-4, 2) B (2, 2)
D (2, -2)
C (-4, -2)
AB = 6 units
DC = 6 units
AC = 4 units BD = 4 units
Opposite sides are equal / congruent.
Rectangle

7. Distance Formula
The distance between two points, whether or not they are aligned
horizontally or vertically, can be determined using the distance formula.
Consider the points M and C whose
coordinates are (x1, y1) and (x2, y2), respectively.
The distance d between these points can be
determined using the distance formula.
(x1, y1) M
(x2, y2) C

8. Distance Formula:
Example 1: Find the distance between M(2, 2) and C(10, 8).
Given: Let M(2, 2) and C(10, 8)
The distance between M and C is 10 units.
(x1, y1) (x2, y2)
M (2, 2)
(10, 8) C

9. Distance Formula:
Example 2: Find the distance between P(7, –3) and S(5, 8).
Given: Let P(7, –3) and S(5, 8)
(x1, y1) (x2, y2)
(5, 8) S
(7, - 3) P

10. Midpoint Formula
(x2, y2) B
A (x1, y1)

11. Midpoint Formula:
Example 1: The coordinates of the endpoints of AB are (-3, -2) and
(5, 4), respectively. What is the coordinates of its midpoint M?
Given: Let A(–3, –2) and B(5, 4)
(x1, y1) (x2, y2)
A (-3, -2)
B (5, 4)
M (1, 1)
The coordinate of the midpoint of AB is (1, 1).

12. Midpoint Formula:
Example 2: The coordinates of the endpoints of LA are (1, -3) and
(6, 3), respectively. What is the coordinates of its midpoint E?
Given: Let L(1, –3) and A(6, 3)
(x1, y1) (x2, y2)
L (1, -3)
A (6, 3)

13. COORDINATE PROOF
A proof that uses figures on a coordinate plane to prove geometric
properties is called a coordinate proof.
To prove geometric properties using the methods of coordinate
geometry, consider the following guidelines for placing figures on a
coordinate plane.
1. Use the origin as vertex or center of a figure.
3. If possible, keep the figure within the first quadrant.
4. Use coordinates that make computations simple and easy. Sometimes, using
coordinates that are multiples of two would make the computation easier.
In some coordinate proofs, the distance formula is applied.

14. Coordinate Proof: Example Problem
Prove that the diagonals of a
rectangle are congruent using the methods
of coordinate geometry.
Given: Quadrilateral LOVE with diagonals
LV and OE.
Prove: LV ≅ OE
1. Use the origin as vertex or center of a figure.
E (0, 0)
2. Place at least one side of a polygon on an axis.
3. If possible, keep the figure within the first quadrant.
4. Use coordinates that make computations simple
and easy. Sometimes, using coordinates that are
multiples of two would make the computation easier.
V (6,0 )
O (6, 4)
L (0, 4)
Question 1. Find the distance between L and V.
Question 2. Find the distance between O and E.

15. Coordinate Proof: Example Problem
Prove that the diagonals of a rectangle are congruent using the methods of coordinate geometry.
Given: Quadrilateral LOVE with diagonals LV and OE.
Prove: LV ≅ OE
E (0, 0) V (6,0)
O (6, 4)
L (0, 4)
Question 1. Find the distance between L and V.
Given: Let L(0, 4) and V(6,0)
Question 2. Find the distance between O and E.
Given: Let O(6, 4) and E(0, 0)
Since LV = 2 √( 13) and OE = 2 √( 13), then LV = OE by
substitution. Therefore, LV ≅ OE. The diagonals of a
rectangle are congruent.

17. Sample Problems:
Applies the distance formula to
prove some geometric
properties.
DIVISION WEBINAR
ON
Enhanced Mathematics
Teaching And Learning
Program (EMTAP)

18. After the discussion, ask the following questions:
1. How does the distance formula help geometric
properties?
2. What are the guidelines to be followed on
coordinate proof to prove geometric properties?

19. Directions: Perform, Analyze and Solve the given problems involving distance formula in proving geometric properties.
1. Triangle PQR with coordinates P(–3, –2), Q(–1, 3), and
R(1, –2). Show that triangle PQR is isosceles triangle.
(-1, 3) Q
(-3,-2) P R (1, -2)
Solution:
PQ = RQ implies PQ ≅ RQ
PQR is isosceles triangle because it has
two congruent sides.

20. Directions: Perform, Analyze and Solve the given problems involving distance formula in proving geometric properties.
1. Quadrilateral MARK has coordinates M(–5, –2),
A(–2, 2), R(2, –1) and K(–1, –5). Show that
diagonal of Quadrilateral MARK is congruent.
Solution:
AK = MR implies AK ≅ MR
The diagonal of the quadrilateral is congruent.
(-2, 2) A
(-5, -2) M
R (2, -1)
(-1, -5) K

21. Directions: Perform, Analyze and Solve the given problems involving distance formula in proving geometric properties.
1. Triangle ART has coordinates A(1, 1), R(–3, 5), and
T(1, 5). Show that triangle ART is isosceles right triangle.
Solution:
(AR)2 = (RT)2 + (TA)2
Pythagorean Theorem
c2 = a2 + b2
16(2) = 16 + 16
The triangle is right because the sides
satisfy the Pythagorean Theorem.
(-3, 5) R T (1, 5)
A (1, 1)
32 = 32
Triangle ART is isosceles
right Triangle.

22. Directions: Perform, Analyze and Solve the given problems involving distance formula in proving geometric properties.
1. Quadrilateral HOPE has coordinates H(–2, 1),
O(3, 6), P(4, –1) and E(–1, –6). Show that
Quadrilateral HOPE is rhombus.
Solution:
(-1, -6) E
(-2, 1) H
O (3, 6)
P (4, -1)

23. Directions: Perform, Analyze and Solve the given problems involving distance formula in proving geometric properties.
1. Quadrilateral HOPE has coordinates H(–2, 1),
O(3, 6), P(4, –1) and E(–1, –6). Show that
Quadrilateral HOPE is rhombus.
Solution:
(-1, -6) E
(-2, 1) H
O (3, 6)
P (4, -1)
HO = OP = PE = EH implies HO ≅ OP ≅ PE ≅ EH
HOPE is a rhombus because all sides are congruent.

24. 1. Define the distance formula and midpoint
formula, then, state the formulas.
LET’S HAVE A DEAL!!!
2. Enumerate the guidelines/steps in using the
coordinate proof.
3. Explain the important role of distance formula
in proving geometric properties.

25. Thank you for your participation and attendance!!!
How did you find the session for today?
What are your experiences in the lesson that
helps you in real life by applying the concepts
you have learned?
CLOSED NA TAYO?

26. Direction: Solve the problem involving the use of distance formula
in proving geometric properties.
A map showing the locations of different municipalities is
drawn on a coordinate plane. Each unit on the coordinate plane is
equivalent to 5 kilometers. Suppose the coordinates of San Mateo
is (2, 2) and Cordon is (5, 8). What is the shortest distance between
these two towns?
1. Show the complete solution of the problem.
2. Submit your output in the google drive link for assignment.
ACTIVITY: I WILL DO IT!!!

27. The capacity to learn is a
gift;
The ability to learn is a
skill;
The willingness to learn is
a choice.
Brian Herbert

28. SALAMUCH PO…